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Mathematical modelling of the unbending of continuously cast steel slabs 1983

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C I MATHEMATICAL MODELLING OF THE UNBENDING OF CONTINUOUSLY CAST STEEL SLABS by MASATSUGU UEHARA M . S . U n i v e r s i t y Of Tokyo,1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES M e t a l l u r g i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA August 1983 © Masatsugu Uehara, 1983 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r he r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d .that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Meta./lu,r^ic6,/ Etplneer'irt^ The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Ouj. /P** , /pr3 ABSTRACT A t w o - d i m e n s i o n a l , e l a s t o - p l a s t i c , f i n i t e - e l e m e n t model has been d e v e l o p e d t o c a l c u l a t e the b e n d i n g and b u l g i n g d e f o r m a t i o n of a p a r t i a l l y s o l i d i f i e d c o n t i n u o u s l y c a s t s t e e l s l a b d u r i n g s t r a i g h t e n i n g on a curved-mould c a s t i n g machine. A p r e l i m i n a r y , t h r e e - d i m e n s i o n a l e l a s t i c a n a l y s i s r e v e a l e d t h a t a t w o - d i m e n s i o n a l p l a n e - s t r e s s model i s s u f f i c i e n t f o r the c a l c u l a t i o n s . The e f f e c t s of s o l i d - s h e l l m o t i o n have been c o n s i d e r e d i n p a r t by s h i f t i n g the r o l l p o i n t s i n two s t e p s . The model was checked by comparing p r e d i c t i o n s of i n t e r n a l c r a c k s w i t h p l a n t d a t a . From t h e r e s u l t s of c a l c u l a t i o n s of a o n e - p o i n t b e n d i n g bow-type c a s t e r ( l 0 . 5 m r a d i u s ) f o r c a s t i n g speeds of 1.0,1.2 and 1.6m/min, i t has been v e r i f i e d t h a t i n t e r n a l c r a c k s appear a t t h e s o l i d i f i c a t i o n f r o n t i n the upper s h e l l due t o s t r a i g h t e n i n g of t h e s t r a n d a t th e h i g h e r c a s t i n g speeds. The c r i t i c a l s t r a i n f o r i n t e r n a l c r a c k s was taken t o be 0.25-0.3% a t a s t r a i n r a t e of 1x10"" s" 1 f o r l o w - c a r b o n s t e e l s . I t has been found t h a t the upper and l o w e r s h e l l s d e f orm s e p a r a t e l y around t h e i r i n d i v i d u a l n e u t r a l a x e s , w h i c h a r e s h i f t e d t o w i t h i n 15mm of the r e s p e c t i v e s o l i d i f i c a t i o n f r o n t s by the r o l l - f r i c t i o n f o r c e . T h e r e f o r e t h e b e n d i n g strain,£ , i n t h e l o w - d u c t i l i t y r e g i o n c l o s e t o t h e s o l i d i f i c a t i o n f r o n t can be v e r y s m a l l , lower by about 0.3% t h a n the v a l u e p r e d i c t e d by one n e u t r a l - a x i s t h e o r y . However, as a r e s u l t of the i n t e r a c t i o n w i t h the b u l g i n g s t r a i n , E B , t h e r e s u l t a n t t o t a l s t r a i n , e x ,becomes l a r g e enough t o c ause i n t e r n a l c r a c k s ( r a d i a l s t r e a k s ) c l o s e t o the s o l i d i f i c a t i o n f r o n t of the upper s h e l l . The c o r r e l a t i o n among t h e s e v a r i a b l e s i s as f o l l o w s ; e T = (2 - 5) e £ + e u Thus, the b u l g i n g s t r a i n a f f e c t s the t o t a l s t r a i n s i g n i f i c a n t l y ; and t o p r e v e n t i n t e r n a l c r a c k s i t i s i m p o r t a n t t o s u p p r e s s t h e b u l g i n g by h a v i n g low s u r f a c e t e m p e r a t u r e s and s m a l l r o l l p i t c h e s d u r i n g s t r a i g h t e n i n g . By comparing machine r a d i i of 8m,10.5m and 13m f o r a o n e - p o i n t b e n d i n g bow-type c a s t e r , i t has been v e r i f i e d t h a t t h e s m a l l machine r a d i u s of 8.0m i s u n f a v o r a b l e because a t normal c a s t i n g speeds t h e t e n s i l e s t r a i n a t the s o l i d i f i c a t i o n f r o n t exceeds th e c r i t i c a l v a l u e f o r c r a c k f o r m a t i o n . iv TABLE OF CONTENTS Page Abstract i i Table of Contents iv L i s t of Tables v i L i s t of Figures v i i L i s t of Symbols x i i Acknowledgement xiv Chapter 1 INTRODUCTION 1 2 PREVIOUS WORK AND OBJECTIVES OF PRESENT WORK .. 4 2.1 Internal cracks in continuously cast slabs . 4 2.2 Previous work on stress analysis of bending and bulging 6 2.3 Objectives of present work 9 3 BENDING/UNBENDING STRESS ANALYSIS OF CONTINUOUSLY CAST SLABS 11 3.1 Introduction 11 3.2 Mechanical properties of low-carbon steels at elevated temperature 13 3.2.1 Types of s t r e s s - s t r a i n curves 14 3.2.2 Mechanical property data 16 3.3 Model development 22 3.3.1 Comparison of the three-dimensional and two-dimensional models 23 3.3.2 E f f e c t s of creep in c a l c u l a t i o n s of bulging 29 3.3.3 Two-dimensional e l a s t o - p l a s t i c F i n i t e ELEMENT 34 3.3.4 Boundary conditions 36 3.3.4.1 Roll f r i c t i o n force" 38 3.3.4.2 Shi f t of the boundary condition . 40 V 3.3.5 C a l c u l a t i o n f l o w 45 4 CALCULATION CONDITIONS 49 5 MODEL PREDICTIONS AND DISCUSSION 63 5.1 R e s u l t s of c a l c u l a t i o n s 63 5.1.1 Comparison of model p r e d i c t i o n w i t h p l a n t d a t a 71 5.1.2 B u l g i n g s t r a i n 73 5.1.3 Bending/Unbending s t r a i n 73 5.2 Corner s t r a i n and c r a c k f o r m a t i o n 88 5.3 Creep e f f e c t s on the c r i t i c a l s t r a i n 94 6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK ... 98 6. 1 C o n c l u s i o n s 98 6.2 S u g g e s t i o n s f o r f u t u r e work 102 REFERENCES 103 A ppendix I M e c h a n i c a l p r o p e r t i e s a d o p t e d i n . t h e b u l g i n g c a l c u l a t i o n f o r c o m p a r i s o n w i t h the e x p e r i m e n t a l r e s u l t s of. M o r i t a 108 II" D e r i v a t i o n of the F i n i t e - E l e m e n t e q u a t i o n s f o r the e l a s t o - p l a s t i c problems 109 I I I M a t e r i a l m a t r i x [D] ( p l a n e s t r e s s ) used i n t h e F i n i t e Element 111 IV T h i c k - w a l l e d c y l i n d e r under i n t e r n a l p r e s s u r e ( p l a n e s t r a i n ) 113 V E s t i m a t i o n of r o l l f r i c t i o n f o r c e i n Case 1 (upper s h e l l ) 117 VI R e s u l t s of c a l c u l a t i o n of bending and b u l g i n g (Case 2 t o Case 10) 118 v i LIST OF TABLES Table Page I Studies of c r i t i c a l s t r a i n for i n t e r n a l cracks ... 5 II Measured data of bulging by M o r i t a 2 6 31 III Calculation conditions for unbending of continuously cast slabs 53 IV Strains at s o l i d i f i c a t i o n front on the center plane normal to the wide face 70 V Maximum bulging d e f l e c t i o n between No.1 and No.2 r o l l s 85 VI Bending and bulging s t r a i n at s o l i d i f i c a t i o n front 95 v i i LIST OF FIGURES Figure Page 1 Schematic drawing of d i f f e r e n t types of casting machines" 0 2 2 Elongation at the s o l i d i f i c a t i o n front during bending 8 3.1 S t r e s s - s t r a i n curves for a u s t e n i t i c iron at elevated temperatures and low s t r a i n r a t e s 1 " 15 3.2 Assumed mechanical properties of sla b at elevated temperature 19 3.3 Strain-hardening exponent as a function of Zener-Hollomon parameter 20 3.4 Influence of strain-hardening exponent on the s t r e s s - s t r a i n curve ' 21 3.5 Schematic diagram of the three-dimensional finite-element mesh for the bending a n a l y s i s 25 3.6 Predicted d i s t o r t i o n s of the slab by the three-dimensional finite-element bending analysis 26 3.7 Predicted xx-strain d i s t r i b u t i o n in the cross section of the slab by the three-dimensional finite-element bending analysis 27 3.8 Comparison of bending s t r a i n s between (a)Three-dimensional and (b)Two-dimensional model 28 3.9 Comparison of maximum bulging predicted by the creep model and e l a s t o - p l a s t i c model 2 2 . 30 3.10 Schematic diagram cf the two-dimensional finite-element mesh for the bulging a n a l y s i s 31 3.11 Comparison of bulging s t r a i n s predicted by the plane stress and plane s t r a i n f i n i t e - element analysis 33 3.12 Influence of the mesh size on bulging s t r a i n in the e l a s t o - p l a s t i c finite-element analysis 33 v i i i Figure Page 3.13 Schematic diagram of the boundary conditions adopted in the two-dimensional finite-element bending analysis 37 3.14 Influence of c o e f f i c i e n t of r o l l f r i c t i o n on the resultant bending str a i n 41 3.15 C o e f f i c i e n t of r o l l f r i c t i o n of hot r o l l i n g as a function of temperature 3 2 42 3.16 Predicted bending s t r a i n with the one-step bending model 43 3.17 Predicted bending and bulging s t r a i n with the one-step bending model 44 3.18 Flow chart for the c a l c u l a t i o n of the bending and bulging s t r a i n 47 3.19 Flow chart for the c a l c u l a t i o n of the bending and bulging s t r a i n ( " e l a s t o - p l a s t i c routine") 48 4.1 Surface temperature and s h e l l thickness in the continuous casting of slab 51 4.2 R o l l p r o f i l e of the 10.5m radius caster .... 52 4.3 R o l l p r o f i l e of the 8.0m radius caster 54 4.4 R o l l p r o f i l e of the 13.0m radius caster .... 55 4.5 Assumed s t r e s s - s t r a i n curves for the slab in Case 1 , 58 4.6 Assumed s t r e s s - s t r a i n curves for the slab in Case 2,3,6,7,8,9 and 10 59 4.7 Assumed s t r e s s - s t r a i n curves for the slab in Case 4 .. 60 4.8 Assumed s t r e s s - s t r a i n curves for the slab in Case 5 .. 61 4.9 Schematic diagram of the two-dimensional finite-element mesh for the bending and bulging analysis 62 i x F i g u r e Page 5.1 P r e d i c t e d d i s t o r t i o n due t o bending and b u l g i n g i n Case 1 64 5.2 P r e d i c t e d x x - s t r a i n c o n t o u r s due t o bending and b u l g i n g i n Case 1 65 5.3 P r e d i c t e d x y - s t r a i n c o n t o u r s due t o bending and b u l g i n g i n Case 1 66 5.4 P r e d i c t e d e f f e c t i v e s t r e s s c o n t o u r s due t o bend i n g and b u l g i n g i n Case 1 67 5.5 P r e d i c t e d p r i n c i p a l s t r a i n v e c t o r s due t o bend i n g and b u l g i n g i n Case 1(upper s h e l l ) . 68 5.6 P r e d i c t e d p r i n c i p a l s t r a i n v e c t o r s due t o bend i n g and b u l g i n g i n Case 1( l o w e r s h e l l ) . 69 5.7 R e l a t i o n between i n t e r n a l c r a c k s and c a s t i n g speed a t O i t a No.4 c a s t e r 3 *' 3 5 72 5.8 S u l f u r p r i n t of a l o n g i t u d i n a l s e c t i o n ; r a t i n g of i n t e r n a l c r a c k s = 0 .2" 6 74 5.9 S u l f u r p r i n t of a l o n g i t u d i n a l s e c t i o n ; r a t i n g of i n t e r n a l c r a c k s =0.5" 6 75 5.10 S u l f u r p r i n t of a l o n g i t u d i n a l s e c t i o n ; r a t i n g of i n t e r n a l cracks=1 . 0* 6 76 5.11 P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , e > i n Case 1 (upper shell,V=1.6m/min) ... . 77 5.12 P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , e x , i n Case 1( l o w e r shell,V=1.6m/min) 78 5.13 P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , E x 11 n Case 2(upper shell,V=1.2m/min) 79 5.14 P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , £ x , i n Case 3(upper shell,V=1.Om/min) 80 5.15 P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , e x , i n Case 3 ( l o w e r shell,V=1.Om/min) 81 5.16 P r e d i c t e d d i s t o r t i o n due t o b u l g i n g i n Case 1 ( l o w e r s h e l l ) 82 X Figure Page 5.17 Predicted x x - s t r a i n due to bulging in Case 1 (lower s h e l l ) ' 83 5.18 Predicted xy-strain due to bulging in Case 1 (lower s h e l l ) 84 5 . 1 9 Predicted bending s t r a i n , e in Case 1(upper shell,V=1.6m/min) 89 5.20 Predicted bending s t r a i n , e in Case H l o w e r shell,V=1.6m/min) 90 5.21 Predicted curvature of the s h e l l due t o bending in Case 1 91 5.22 Relation between bending s t r a i n , E x , a n d r o l l p i t c h predicted by the finite-element bending analys i s 92 5.23 Relation between bending s t r a i n , e x , a n d s h e l l thickness predicted by the finite-element bending analysis(Machine radius=10.5m) 92 5.24 Relation between bending s t r a i n , e x , a n d s h e l l thickness predicted by the finite-element bending analys is(Machine radius=8.0m) 93 5.25 Relation between bending s t r a i n , e ,and machine radius(curvature) predicted by the finite-element bending analysis 93 5.26 Predicted t o t a l bending and bulging s t r a i n , E j , a t an inner surface as a f u n c t i o n , of bulging s t r a i n , e B ,and bending s t r a i n , e ^ .... 97 IV.1 Geometry of a thick walled cylinder 115 IV.2 Comparison of the calcul a t e d stresses 0„(solid points and l i n e s ) with those obtained by H i l l 3 0 , 116 VI.1 Predicted bending s t r a i n , £ ,in Case 2 (upper sh e l l ) ? 119 VI.2 Predicted bending s t r a i n , e x , i n Case 3 (upper shell) 120 x i F i g u r e Page V I . 3 P r e d i c t e d b e n d i n g s t r a i n , e , i n Case 3 ( l o w e r s h e l l ) 121 VI.4 P r e d i c t e d c u r v a t u r e of t h e s h e l l due t o bend i n g i n Case 3 122 VI.5 P r e d i c t e d b e n d i n g ' s t r a i n , e , i n Case 4 (upper s h e l l ) 123 VI.6 P r e d i c t e d bending and b u l g i n g s t r a i n , z , i n Case 4(upper s h e l l ) 124 VI.7 P r e d i c t e d bending s t r a i n , E , i n Case 5 (upper s h e l l ) .x 125 VI.8 P r e d i c t e d bending and b u l g i n g strain,£ , i n Case 5(upper s h e l l ) x 126 VI.9 P r e d i c t e d bending s t r a i n , e , i n Case 6 (upper s h e l l ) x 127 VI.10 P r e d i c t e d bending and b u l g i n g s t r a i n , e , i n Case 6(upper s h e l l ) x 128 VI.11 P r e d i c t e d bending s t r a i n , e , i n Case 7 (upper s h e l l ) ..x 129 VI.12 P r e d i c t e d bending and b u l g i n g s t r a i n , E , i n Case 7(upper s h e l l ) • .. . x 130 VI.13 P r e d i c t e d bending s t r a i n , e , i n Case 8 (upper s h e l l ) .x 131 VI.14 P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , e , i n Case 8 (upper s h e l l ) . x 132 VI.15 P r e d i c t e d bending s t r a i n , e , i n Case 9 (upper s h e l l ) x 133 VI.16 P r e d i c t e d b e n d i n g and b u l g i n g s t r a i n , E , i n Case 9(upper s h e l l ) x 134 VI.17 P r e d i c t e d bending s t r a i n , E , i n Case 10 (upper s h e l l ) • .x 135 VI.18 P r e d i c t e d bending and b u l g i n g s t r a i n , e , i n Case I0( u p p e r s h e l l ) ? 136 x i i L IST OF SYMBOLS d s l a b t h i c k n e s s (mm) d 0 g r a i n s i z e (y m) [D] m a t e r i a l m a t r i x E Young's modulus (MPa) F r o l l f r i c t i o n f o r c e (N) i [K] s t i f f n e s s m a t r i x 1 r o l l p i t c h ( m m ) Ri * A l e l o n g a t i o n (mm) n s t r a i n - h a r d e n i n g exponent N number of r o l l s n e c e s s a r y t o a b s o r b the b e n d i n g d e f o r m a t i o n [N] m a t r i x of shape f u n c t i o n p f e r r o s t a t i c p r e s s u r e (MPa) i Q s e l f - d i f f u s i o n energy (J/mol) R machine r a d i u s (mm) R 0 gas c o n s t a n t (J/mol°K) s s h e l l t h i c k n e s s (mm) T t e m p e r a t u r e (°C) T 0 s u r f a c e t e m p e r a t u r e (°C) "u x a v e r a g e d i s p l a c e m e n t i n x d i r e c t i o n (mm) u Y d i s p l a c e m e n t i n y d i r e c t i o n (mm) V c a s t i n g speed (m/min) w s l a b w i d t h (mm) A y d i s t a n c e between o u t e r and i n n e r s u r f a c e s (mm) y d i s t a n c e from the n e u t r a l a x i s (mm) Zener-Hollomon parameter ( s ~ 1 ) maximum b u l g i n g d e f l e c t i o n (mm) s t r a i n s t r a i n at i n n e r s u r f a c e s t r a i n at o u t e r s u r f a c e s t r a i n r a t e ( s ~ 1 ) be n d i n g s t r a i n b u l g i n g s t r a i n t o t a l of b e n d i n g and b u l g i n g s t r a i n e f f e c t i v e p l a s t i c s t r a i n c u r v a t u r e of the s t r a n d (1/mm) s t r e s s (MPa) e f f e c t i v e s t r e s s (MPa) peak s t r e s s (MPa) Y i e l d s t r e s s (MPa) f r i c t i o n a l c o e f f i c i e n t P o i s s o n ' s r a t i o x i v A C K N O W L E D G E M E N T S I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e t o Dr.J.K.Brimacombe and Dr. I . V . S a m a r a s e k e r a f o r t h e i r u s e f u l a d v i c e and g u i d a n c e t h r o u g h o u t the c o u r s e of t h i s s t u d y . Thanks a r e a l s o e x t e n d e d t o my f e l l o w g r a d u a t e s t u d e n t s and f a c u l t y members i n the Department of M e t a l l u r g i c a l E n g i n e e r i n g . The a s s i s t a n c e of the t e c h n i c a l s t a f f and i n p a r t i c u l a r t h a t of Mr.N.Walker i s g r e a t l y a p p r e c i a t e d . I am a l s o g r a t e f u l t o the N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada and Nippon S t e e l Corp. i n Japan f o r p r o v i d i n g f i n a n c i a l s u p p o r t . S p e c i a l thanks must a l s o be g i v e n t o Mr.H.Misumi f o r h i s a s s i s t a n c e i n p r o v i d i n g the p l a n t d a t a . F i n a l l y , I would l i k e t o tak e t h i s o p p o r t u n i t y t o thank my w i f e Teruko f o r her a s s i s t a n c e i n t h i s work. 1 C h a p t e r . 1 INTRODUCTION D u r i n g the l a s t two decades s e v e r a l t y p e s of c o n t i n u o u s - c a s t i n g m a c h i n e s have been d e v e l o p e d f o r t h e p r o d u c t i o n of s t e e l s l a b s . The machine t y p e s i n c l u d e t h e v e r t i c a l , v e r t i c a l w i t h b e n d i n g , c i r c u l a r a r c and o v a l bow which a r e shown s c h e m a t i c a l l y i n F i g . 1 . W i t h i n c r e a s i n g demands on p r o d u c t i o n r a t e and p r o d u c t q u a l i t y the t r e n d has been away from t h e v e r t i c a l t y p e t o t h e low-head,'bow-type c a s t e r . I n t h e case o f the l a t t e r , however, the s t r a n d i s s t r a i g h t e n e d w h i l e c o n t a i n i n g a l i g u i d c o r e . The r e s u l t a n t b e n d i n g g i v e s r i s e t o e l o n g a t i o n , w h i c h can cause i n t e r n a l c r a c k s , c l o s e . t o t h e s o l i d i f i c a t i o n f r o n t where t h e d u c t i l i t y of t h e s t e e l i s v e r y low. I n a d d i t i o n o t h e r t y p e s of s t r e s s e s a r e a p p l i e d to t h e s o l i d i f y i n g s h e l l , v i z . b u l g i n g s t r e s s e s due t o f e r r o s t a t i c p r e s s u r e and t h e r m a l s t r e s s e s . 1 ' 2 ' 1 6 Even t h o u g h t h e f e r r o s t a i c p r e s s u r e i s r e l a t i v e l y s m a l l i n a low head bow-type c a s t e r , b u l g i n g s t r a i n may combine w i t h b e n d i n g s t r a i n t o c r e a t e e x c e s s i v e s t r a i n s a t t h e s o l i d i f i c a t i o n f r o n t . T h i s e f f e c t w i l l v e r t i c a l t y p e v e r t i c a l t y p e c i r c u l a r a r c t y p e o v a l b o w t y p e w i t h b e n d i n g ( o n e - p o i n t b e n d i n g ) ( m u l t i - p o i n t b e n d i n g ) F i g . l Schematic drawing of Different Types of Casting Machines^? 3 be shown i n t h i s t h e s i s . As i n c r e a s i n g emphasis has been p l a c e d on p r o d u c t q u a l i t y , s t r e s s a n a l y s i s has been one of t h e t o o l s employed t o stu d y t h e f o r m a t i o n of i n t e r n a l c r a c k s i n c o n t i n u o u s l y c a s t s t e e l s s i n c e t h e l a t t e r h a l f o f the 1970's. However, o n l y a few s t u d i e s have been r e p o r t e d on the a n a l y s i s of s t r e s s e s i n bend i n g ; and hence t h e t a s k i s by no means c o m p l e t e . F u r t h e r i n v e s t i g a t i o n s a r e u r g e n t l y r e q u i r e d on t h i s s u b j e c t from a d e s i g n p o i n t of v i e w , because optimum and l i m i t i n g machine d e s i g n s a r e r e q u i r e d i n o r d e r t o meet the s t r i c t demands f o r the modern low head c a s t e r s . In the p r e s e n t work, the s t r e s s a n a l y s i s of o n e - p o i n t bending w i l l be c o n s i d e r e d as a f i r s t s t e p t o w a r d a b e t t e r u n d e r s t a n d i n g of b e n d i n g b e h a v i o r . 4 Chapter 2 PREVIOUS WORK AND OBJECTIVES OF PRESENT WORK 2.1 I n t e r n a l c r a c k s i n c o n t i n u o u s l y c a s t s l a b s I t has l o n g been r e c o g n i z e d t h a t even s m a l l t e n s i l e s t r a i n s a p p l i e d t o the s o l i d i f y i n g s t e e l s h e l l can l e a d t o the g e n e r a t i o n o f i n t e r n a l c r a c k s ( s o l i d i f i c a t i o n c r a c k s ) c l o s e t o the s o l i d i f i c a t i o n f r o n t . " 1 ' " 2 The i n t e r n a l c r a c k s due t o b u l g i n g or bendi n g of s l a b s can be seen normal t o the broad f a c e i n a l o n g i t u d i n a l s e c t i o n . I n most c a s e s , b e n d i n g c r a c k s a r e formed i n the upper s h e l l i n t h e s t r a i g h t e n i n g zone and b u l g i n g c r a c k s a r e g e n e r a t e d i n upper and lo w e r s h e l l s beneath the r o l l s u p p o r t p o i n t s . These c r a c k s a r e v i s i b l e on s u l p h u r p r i n t s s i n c e t h e y a r e g e n e r a l l y f i l l e d w i t h s o l u t e r i c h r e s i d u a l l i q u i d (see Figs.5.8 - 5.10 S e c t i o n 5.1.1). The c r i t i c a l s t r a i n s f o r i n t e r n a l c r a c k s have been r e p o r t e d t o be dependent on s t e e l grades and s t r a i n r a t e s w i t h l o w - c a r b o n s t e e l s and low s t r a i n r a t e s g i v i n g h i g h c r i t i c a l s t r a i n s . T a b l e I p r e s e n t s r e p o r t e d v a l u e s of c r i t i c a l s t r a i n f o r t h e s e c o n d i t i o n s . Some s c a t t e r i n the data'(0.2 - 3.0%) can 5 T a b l e I S t u d i e s o f c r i t i c a l s t r a i n f o r i n t e r n a l c r a c k s S t u d y M e t h o d u s e d t o o b t a i n c r i t i c a l s t r a i n C r i t i c a l s t r a i n c r i t R ef P a l m a e r s B a s e d on e l a s t o - p l a s t i c t h e r m a l s t r e s s c a l c u l a t i o n o f a c o n t i n u o u s l y c a s t b l o o m . C=.18% e ? 0.2% 1 2 P u h r i n g e r B a s e d on e l a s t o - p l a s t i c - c r e e p b u l g i n g c a l c u l a t i o n o f a c o n t i n u o u s l y c a s t s l a b . C=.05% e =6x10-« s" 1 0.39% 15 D a n i e l B a s e d on e l a s t o - p l a s t i c b u l g i n g c a l c u l a t i o n o f a c o n t i n u o u s l y c a s t ' b l o o m . C=.15% e ? 0.6% 36 N a r i t a R o l l m i s a l i g n m e n t t e s t on a c o n t i n u o u s l y c a s t b l o o m . S t r a i n i s c a l c u l a t e d b a s e d on e l a s t o - p l a s t i c s i m u l a t i o n m o d e l . C=.15% e =4x10-" s " 1 0.3% 37 S u z u k i R e d u c t i o n t e s t . Low c a r b o n s t e e l . E =1x10"* s " 1 0.25-0.6% 38 M a t s u m i y a L a b o r a t o r y 3 - p o i n t b e n d i n g t e s t . S p e c i m e n i s p a r t l y m e l t e d by e l e c t r i c a l i n p u t . C = .15% e =5x10-* s" 1 2.0-3.0% 39 6 be seen owing t o t h e d i f f e r e n t methods used t o e s t i m a t e t h e c r i t i c a l s t r a i n a t the s o l i d i f i c a t i o n f r o n t . S i n c e i t i s d i f f i c u l t t o measure the s t r a i n a t t h i s l o c a t i o n , t h e . c r i t i c a l v a l u e has been c a l c u l a t e d i n t h e s e s t u d i e s w i t h some e x p e r i m e n t a l i n p u t . From T a b l e I , f o u r of t h e s i x a u t h o r s r e p o r t a v a l u e i n the range of 0.2-0.5% ;and t h i s has been a d o p t e d i n t h e p r e s e n t s t u d y . To overcome the p r o b l e m of i n t e r n a l c r a c k s due t o s t r a i g h t e n i n g , the f o l l o w i n g two s t e p s have been a d o p t e d i n p r a c t i c e : " c o m p r e s s i o n c a s t i n g " and " m u l t i - p o i n t b e n d i n g " . The p r i n c i p l e of the c o m p r e s s i o n c a s t i n g method i s t o c a n c e l h a r m f u l t e n s i l e s t r e s s e s a t t h e s o l i d i f i c a t i o n f r o n t by a p p l y i n g a c o m p r e s s i v e f o r c e i n t h e b e n d i n g zone. To t h i s end, s e v e r a l d r i v e n r o l l s are r e q u i r e d b e f o r e the b e n d i n g p o i n t t o push t h e s t r a n d w i t h a l a r g e f o r c e a g a i n s t s e v e r a l r o l l s f o l l o w i n g the b e n d i n g p o i n t whose movement i s e l e c t r i c a l l y c o n t r o l l e d so as t o a p p l y a b r a k i n g f o r c e t o t h e s t r a n d and , a t the same t i m e , p r e v e n t s l a b s l i p p a g e . 3 " In t h e c a s e of m u l t i - p o i n t b e n d i n g the s t r a i g h t e n i n g i s d i v i d e d i n t o s e v e r a l b e n d i n g s t e p s and hence t e n s i l e s t r e s s e s a t the s o l i d i f i c a t i o n f r o n t a r e g r e a t l y r e d u c e d . * 3 M u l t i - p o i n t b e n d i n g c a s t e r s a r e under development a t p r e s e n t as a new t ype of low-head, o v a l - bow machine. 2.2 P r e v i o u s work on s t r e s s a n a l y s i s of b e n d i n g and b u l g i n g 7 As mentioned previously, owing to the i n t e r a c t i o n of bending s t r a i n and bulging s t r a i n i n t e r n a l cracks can occur very e a s i l y in the straightening zone in a bow-type caster. Bulging and bending must be analysed simultaneously to study the formation of int e r n a l cracks due to unbending. Numerous studies have been devoted to bulging analysis 1 2 1 , 5 ' 2 0 " 2 7 ; the r e s u l t s w i l l be discussed in d e t a i l in Section 3.3.2. However only a few mathematical models have been reported on bending 1 5 ' 3 1 ' * *'" 5 and no model has yet been reported in the l i t e r a t u r e on the bulging and bending analysis of continuously cast slabs which i s the subject of the present t h e s i s . The s i n g l e beam theory, which assumes a neutral axis at the center of the slab thickness, has long been used to explain the bending behavior of continuously cast slabs. According to this beam theory, the bending s t r a i n s , c u , for one-point and multi-point bending,shown schematically in Fig.2, are given respectively by the following equations . one-point bend ; E U = (-— - s) / R0 O) multi-point bend ; e = - s ) (I/R - I / R ) (2) u I n n-1 n As i s apparent from E q . ( l ) , the bending s t r a i n becomes larger with decreasing s h e l l thickness and machine radius. However,the assumption of one neutral axis i s questionable at the center of Compression! (a) one-point bending (b) mul t i-po in t bending Fig. 2 Elongation at the Solidification Front during Bending. 00 9 a wide f a c e of a s l a b . Based on t h r e e - d i m e n s i o n a l e l a s t i c a n a l y s i s V a t e r l a u s ' 4 has proposed a two n e u t r a l - a x e s t h e o r y i n w hich the upper and l o w e r s h e l l s deform i n d e p e n d e n t l y . However h i s model p r e d i c t i o n s a r e c o n t r a d i c t o r y t o the o b s e r v a t i o n s of i n t e r n a l c r a c k s , i . e . the h a r m f u l t e n s i l e s t r a i n s due t o unbending o c c u r a t the s o l i d i f i c a t i o n f r o n t i n the lower s h e l l and hence i n t e r n a l c r a c k s must appear i n the l ower s h e l l . A dynamic s i m u l a t i o n model has been r e p o r t e d by O n i s h i 3 1 based on the o n e - d i m e n s i o n a l , e l a s t o - p l a s t i c , f i n i t e - e l e m e n t method. U n f o r t u n a t e l y t h i s model i s o n l y s u i t a b l e f o r c a l c u l a t i o n of r o l l r e a c t i o n f o r c e s , and cannot be used t o e v a l u a t e s t r a i n d i s t r i b u t i o n s t h r o u g h the s h e l l t h i c k n e s s because i t i s based on an a s s u m p t i o n of one n e u t r a l - a x i s t h e o r y . 2.3 O b j e c t i v e s of p r e s e n t work The p r e s e n t s t u d y has been u n d e r t a k e n t o e l u c i d a t e machine d e s i g n p a r a m e t e r s and c a s t i n g c o n d i t i o n s t h a t have a s t r o n g i n f l u e n c e on t h e s t a t e of s t r a i n and on c r a c k f o r m a t i o n d u r i n g the unbending of p a r t i a l l y s o l i d i f i e d s t e e l s l a b s . A t w o - d i m e n s i o n a l , e l a s t o - p l a s t i c , f i n i t e - e l e m e n t model has been f o r m u l a t e d f o r t h i s purpose . The p r i m a r y c o n c e r n s of the a n a l y s i s a r e as f o l l o w s : ' ( 1 ) To d e t e r m i n e the unbending b e h a v i o r , i . e . t o a s c e r t a i n whether the c o n v e n t i o n a l s i n g l e n e u t r a l - a x i s t h e o r y i s c o r r e c t or n o t . 10 (2) To c a l c u l a t e the c r i t i c a l s t r a i n f o r i n t e r n a l c r a c k s i n unbending and t o compare i t w i t h the v a l u e s r e p o r t e d i n the l i t e r a t u r e . (3) To f i n d a c o r r e l a t i o n between r e s u l t a n t t o t a l s t r a i n , e , and each of t h e components of b u l g i n g s t r a i n , E , T B and bending s t r a i n , e u The p r e s e n t o n e - p o i n t bending model w i l l p r o v i d e the b a s i s f o r the d e s i g n of a new low-head,bow-type c a s t e r . 11 C h a p t e r 3 BENDING/UNBENDING STRESS ANALYSIS OF CONTINUOUSLY CAST SLABS 3 . 1 I n t r o d u c t i o n S t r e s s a n a l y s i s has been p e r f o r m e d of t h e b e n d i n g / u n b e n d i n g of p a r t i a l l y . s o l i d i f i e d wide s t e e l s l a b s . T h i s i s a v e r y c o m p l i c a t e d p r o b l e m because w h i l e p a s s i n g t h r o u g h t h e s t r a i g h t e n i n g zone the s t r a n d i s s u b j e c t e d a l t e r n a t e l y t o t e n s i o n and c o m p r e s s i o n due t o t h e i n t e r a c t i o n of f e r r o s t a t i c p r e s s u r e p u s h i n g t h e s o l i d i f i e d s h e l l o u t w a r d and t h e r o l l s p u s h i n g a g a i n s t the s h e l l i n t h e o p p o s i t e d i r e c t i o n . Thus each element of t h e s t r a n d e x h i b i t s a c o m p l e x h y s t e r e s i s . A t t h e c e n t e r p l a n e of t h e wide f a c e of t h e s l a b , d e f o r m a t i o n o f t h e s t r a n d i s enhanced as a r e s u l t of i n t e r a c t i o n between b e n d i n g and b u l g i n g . On t h e o t h e r hand a t t h e c o r n e r , d e f o r m a t i o n i s p r i m a r i l y due t o b e n d i n g . T h us, i n f o r m u l a t i n g a model t o c a l c u l a t e b e n d i n g o f t h e moving s t r a n d w i t h a l i q u i d c o r e , b e n d i n g and b u l g i n g d e f o r m a t i o n s have t o be c o n s i d e r e d s i m u l t a n e o u s l y as a t h r e e - d i m e n s i o n a l , v i s c o - e l a s t i c - p l a s t i c p r o b l e m . T h i s i n t r o d u c e s c o n s i d e r a b l e c o m p l e x i t y t o t h e p r o b l e m and can r e s u l t i n p r o h i b i t i v e l y h i g h c o m p u t i n g c o s t s . 12 I n o r d e r t o r e n d e r the problem i n t o a more t r a c t a b l e form the f o l l o w i n g s t e p s were t a k e n . F i r s t l y , t h r e e - d i m e n s i o n a l e l a s t i c a n a l y s i s was a p p l i e d t o the b e n d i n g of the wide s l a b , where t h e s t r a n d was c o n s i d e r e d t o be a h o l l o w box w i t h t e m p e r a t u r e g r a d i e n t s t h r o u g h t h e s h e l l t h i c k n e s s . R e s u l t s of t h e c a l c u l a t i o n have shown t h a t the d e f o r m a t i o n a t the c e n t e r p l a n e of t h e wide f a c e i s independent of t h e s i d e edge. Comparison of t h i s r e s u l t w i t h t h a t of a t w o - d i m e n s i o n a l p l a n e - s t r e s s model of the c e n t e r p l a n e o f a s l a b has i n d i c a t e d t h a t a t w o - d i m e n s i o n a l model can be a p p l i e d t o the bending a n a l y s i s of wide s l a b s as i n t h e case of b u l g i n g a n a l y s i s . 2 1 Thus, a t w o - d i m e n s i o n a l model has been f o r m u l a t e d f o r t h e l o n g i t u d i n a l s e c t i o n a t t h e c e n t e r p l a n e of t h e wide f a c e of s l a b . In the f o r m u l a t i o n of t h e model e l a s t o - p l a s t i c b e h a v i o r has been i n c o r p o r a t e d but the e f f e c t s of c r e e p were not c o n s i d e r e d . . D u r i n g s t r a i g h t e n i n g , the p r i n c i p a l component of the t o t a l s t r a i n i s r e l a t e d t o e l a s t o - p l a s t i c d e f o r m a t i o n w h i l e f o r b u l g i n g i f the r o l l s p a c i n g i s s u f f i c i e n t l y s m a l l c r e e p c a n be n e g l i g i b l e . 2 2 S i n c e d i s p l a c e m e n t boundary c o n d i t i o n s a r e imposed, i t i s a n t i c i p a t e d t h a t c r e e p w i l l not enhance t h e t o t a l s t r a i n but w i l l cause a s t r e s s r e l a x a t i o n i n s t e a d i n the b e n d i n g a n a l y s i s . For the boundary c o n d i t i o n s , which a r e u s u a l l y t h e most i m p o r t a n t a s p e c t o f a m a t h e m a t i c a l model, t h e f o l l o w i n g two f a c t o r s s p e c i f i c a l l y have been c o n s i d e r e d . The c a l c u l a t i o n 1 3 has been p e r f o r m e d i n two s t a g e s and f o r each s t a g e th e r o l l s u p p o r t s have been a p p r o p r i a t e l y chosen t o s i m u l a t e as s i m p l y as p o s s i b l e a moving s t r a n d . T h i s semi-dynamic s i m u l a t i o n p r o v e d u s e f u l p a r t i c u l a r l y f o r the l ower s h e l l and r e s u l t e d i n a smoother s t r a i n d i s t r i b u t i o n a t the s o l i d - l i q u i d i n t e r f a c e . S e c o n d l y , r o l l f r i c t i o n f o r c e has been c o n s i d e r e d . T h i s a p p r o a c h i s d i f f e r e n t from the u s u a l l y a d o p t e d c o n c e p t of w i t h d r a w a l r e s i s t a n c e i n t h a t t h i s f o r c e i s d e r i v e d from the b e n d i n g d e f o r m a t i o n of the s l a b . T h i s r o l l f r i c t i o n f o r c e has been used as the f o r c e boundary c o n d i t i o n on the upstream edge of t h e s h e l l i n the f i n i t e - e l e m e n t a n a l y s i s . These t o p i c s a r e d i s c u s s e d i n g r e a t e r d e t a i l i n the subsequent s e c t i o n s . 3.2 M e c h a n i c a l p r o p e r t i e s of low c a r b o n s t e e l s a t e l e v a t e d t e m p e r a t u r e A f a c t o r of paramount i m p o r t a n c e i n a m o d e l l i n g s t u d y of t h i s k i n d i s t h e a c c u r a c y of the m e c h a n i c a l p r o p e r t y d a t a . The m e c h a n i c a l p r o p e r t i e s of s t e e l a t e l e v a t e d t e m p e r a t u r e s a r e dependent on t e m p e r a t u r e , s t r a i n r a t e , t h e r m a l h i s t o r y , s t r u c t u r e and c h e m i c a l c o m p o s i t i o n . In o r d e r t o c a l c u l a t e the s t r e s s e s i n the s o l i d i f y i n g s h e l l of c o n t i n u o u s l y c a s t s l a b s the p r o p e r t y d a t a u t i l i z e d s h o u l d be o b t a i n e d from t e s t s c o n d u c t e d under c o n d i t i o n s s i m i l a r t o t h o s e i n a c a s t e r . In the p a s t few y e a r s s e v e r a l s t u d i e s have been c o n d u c t e d t o d e t e r m i n e th e p l a s t i c b e h a v i o r of s t e e l s a t e l e v a t e d t e m p e r a t u r e s f o r a v a r i e t y of s t r a i n r a t e s . 3 " 1 6 S i n c e 14 a t t h e s e t e m p e r a t u r e s i t i s d i f f i c u l t t o s e p a r a t e out the e f f e c t s of c r e e p from the d a t a i t i s i m p o r t a n t t o choose d a t a o b t a i n e d f o r s t r a i n r a t e s comparable t o t h o s e e n c o u n t e r e d i n c o n t i n u o u s c a s t i n g . I t i s t h u s p o s s i b l e t o p a r t i a l l y a ccount f o r t h e e f f e c t s of c r e e p i n m o d e l l i n g the s t r a i n d i s t r i b u t i o n i n t h e s t r a n d , w i t h an e l a s t o - p l a s t i c model. For t h i s s t u d y , m e c h a n i c a l p r o p e r t i e s of s t e e l i n the t e m p e r a t u r e range 900°C t o the s o l i d u s t e m p e r a t u r e and f o r s t r a i n r a t e s i n the range from 10" 5 t o 10~ 2 s " 1 must be known. P r o p e r t i e s p a r t i c u l a r l y i m p o r t a n t are 1) Young's modulus,E 2) Y i e l d s t r e s s , a ^ 3) P o i s s o n ' s r a t i o , v 4) S t r a i n - h a r d e n i n g exponent,n The t e m p e r a t u r e s and s t r a i n r a t e s g i v e n above a r e t y p i c a l of v a l u e s e n c o u n t e r e d i n c o n t i n u o u s c a s t i n g . 3 3.2.1 Types of s t r e s s - s t r a i n c u r v e s B e f o r e p r o c e e d i n g t o the m e c h a n i c a l p r o p e r t y d a t a , i t i s i m p o r t a n t t o u n d e r s t a n d t h e g e n e r a l f e a t u r e s of t h e f l o w c u r v e s a t h i g h t e m p e r a t u r e s and low s t r a i n r a t e s . Three t y p e s of s t r e s s - s t r a i n c u r v e s have been r e p o r t e d f o r a u s t e n i t i c i r o n a t e l e v a t e d t e m p e r a t u r e s and low s t r a i n r a t e s , 1 1 ' 1 * as shown i n F i g . 3 . 1 1" . Type-1; The f l o w s t r e s s i n c r e a s e s t o a peak v a l u e 15 C . % T'C(x100) 13 12 11 10 9 325 8 775 0.10 E=^7x1CJ4/s • • • A o • x10"3/s • • A A • * *10~4S • • A • • 0.25 E=6.7><10"Vc; • • • • A • • x i a 3 / s • • • A • D • x10"2/S A A • • D 0.39 e=5.7x10"44 • • • • A AO • * x10"3/s • A A • — • • x10'2/s A A A • • — • 0.71 £=6.7*104/s • • • C A A • • • * x ia 3 / s A A A A • — • — " *10"£S A A A A • — • — • T Y P E - t A T Y P E - 2 aTYPE-3 Fig. 3.1 Stress-Strain Curves for Austenitic Iron at Elevated Temperatures and Low Strain Rates.^ 16 a n d t h e n f a l l s t o a l e v e l w h i c h o s c i l l a t e s a b o u t a mean. ( a t h i g h t e m p e r a t u r e s and low s t r a i n r a t e s ) T y p e - 2 ; The f l o w s t r e s s i n c r e a s e s t o a p e a k v a l u e a n d t h e n f a l l s t o a s t e a d y - s t a t e l e v e l . ( b e t w e e n v a l u e s o f Type-1 a n d 3) T y p e - 3 ; The f l o w s t r e s s i n c r e a s e s t o a maximum v a l u e a n d t h e n d e c r e a s e s r a p i d l y w i t h o u t r e a c h i n g a s t e a d y - s t a t e . ( a t low t e m p e r a t u r e s a n d h i g h s t r a i n r a t e s ) The s t r e s s - s t r a i n d a t a r e q u i r e d f o r t h e p r e s e n t s t u d y i s f o r t h e i n i t i a l s t r a i n h a r d e n i n g r e g i o n , t h a t i s up t o a b o u t 1% o f t h e t r u e s t r a i n . 3.2.2 M e c h a n i c a l p r o p e r t y d a t a The f o l l o w i n g d a t a h a v e b e e n u s e d i n t h i s w o r k . 1) Y o u n g ' s m o d u l u s , E The d a t a o b t a i n e d by M i z u k a m i 7 f o r a 0.08%C s t e e l , f r o m t e n s i l e t e s t s and a r e s o n a n c e m e t h o d , were a d o p t e d a s shown i n F i g . 3 . 2 . The r e s u l t s o f t h e t e n s i l e t e s t s show t h a t t h e Y o u n g ' s m o d u l u s i s i n d e p e n d e n t o f s t r a i n r a t e s i n t h e r e g i o n f r o m 1 x 1 0 " " t o 3 x 1 0 " 3 s " 1 . - T h e d e p e n d e n c e o f E on t e m p e r a t u r e i s a s f o l l o w s . 17 1 0 0 0 < T < 1 4 0 0 ° C E = 1 . 9 6 X 1 0 " - 1 8 . 3 7 5 ( T - 1 0 0 0 ) MPa ( 3 ) 1 4 0 0 < T < 1 4 7 5 E=1 . 2 2 5 X 1 0 " ( 1 4 7 5 - T ) / 7 5 MPa ( 4 ) T > 1 4 7 5 E = 0 MPa ( 5 ) 2 ) Y i e l d s t r e s s , „ Y The d a t a o b t a i n e d by N i e d e r m a y r 1 6 a t low s t r a i n r a t e s were employed as shown in F i g . 3 . 2 . F o r the h i g h e r s t r a i n r a t e s above 10" 2 s~ 1 , a h i g h e r y i e l d s t r e s s has been o b s e r v e d by J o n a s 5 . The f o r m u l a t i o n of y i e l d s t r e s s by N iedermayr i a as f o l l o w s . 1000<T<1200°C o y = 6 6 . 1 5 - 4 . 6 5 5X1 0 " 2T MPa (6) 1 2 0 0 < T < 1 4 8 0 T > 1 4 8 0 O Y = 5 4 . 3 9 - 3 . 6 7 5 X 1 0 - 2 T MPa ( 7 ) a y = 0 MPa (8) 3 ) P o i s s o n ' s r a t i o , v P o i s s o n ' s r a t i o was assumed to be t e m p e r a t u r e d e p e n d e n t 1 7 ' 1 8 , as shown i n F i g . 3 . 2 . v =8 .23x10 ' 5 T+0.278 ( 9 ) 18 4) S tra in-hardening exponent, n The fo l lowing s t r e s s - s t r a i n r e l a t i o n s h i p was used to simulate the p l a s t i c i t y . where K is a constant , E is the true s t r a i n , and n is the s tra in-hardening exponent. The exponent n depends in a complex way on such parameters as temperature, s t r a i n rate , t o t a l s t r a i n , grain s i z e , e t c . , and therefore cannot be expressed by a simple equation. However, a c o r r e l a t i o n has been observed recent ly by S a k a i 6 between the s tra in-hardening exponent and the f i r s t peak stress a_. or hence Zener-Hollomon parameter,Z, under condit ions of c o n t r o l l e d grain s ize (d0=38 m̂ and 42.3 Pm) as shown in F i g . 3 . 3 . The Zener-Hollomon parameter is given as fo l lows . a = K n E (10) Z= £ exp(- Q s- 1 (11) R T o where, Q=(se l f -d i f fus ion energy) J/mol R=8.319 (gas constant) J / m o l ° K and A,m are constants . The value of n in t h eY -phase increases monotonical ly with increas ing peak s tress » ° >or Z. The data c a l c u l a t e d from P a l m a e r s 1 2 , I I I I I L o o o o o o • • • • • - » N ) U ) 4> CJ1 Poisson's r a t i o , y 0.6 c CD c o CL X cu cn c 0.5 h OA c 0 3 T J o c o -4—» LO 0.2 0.1 0 S a k a i 6 0.036%C S a k a i 6 0.16 % C d0=42.3urn - Palmaers 1 2 0.088%C d 0 = 3 8 u m X • ' ^— —• .1 . 1 . i i . i i 10' 10' 10 8 10 10 10 12 10 Sec -1 Q Zener-Hollomon parameter , Z ( =£ exp (-5-7-) ) Fig. 3.3 Strain-Hardening Exponent as a Function of Zener-Hollomon Parameter, o 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 S t r a i n (%) Fig. 3.4 Influence of Strain-Hardening Exponent on the Stress-Strain Curve. 22 a l t h o u g h t h e g r a i n s i z e i s unknown, a l s o s h ows a m o n o t o n i c i n c r e a s e o f n a n d i s i n g o o d a g r e e m e n t w i t h t h a t o f S a k a i , .as shown i n . F i g . 3 . 3 . T h u s f o r t h e p r e s e n t s t u d y , t h e s t r a i n h a r d e n i n g e x p o n e n t n was t a k e n f r o m F i g . 3 . 3 by c a l c u l a t i n g t h e c o r r e s p o n d i n g Z e n e r - H o l l o m o n p a r a m e t e r Z. The p l o t t e d d a t a o f n shows some s c a t t e r , o f a b o u t 0.05 i n F i g . 3 . 3 b u t t h i s i s u n i m p o r t a n t when v i e w e d i n t e r m s o f t h e r e s u l t a n t s t r e s s - s t r a i n c u r v e s , a s shown i n F i g . 3 . 4 . 3.3 M o d e l d e v e l o p m e n t Owing t o t h e c o m p l e x i t y of t h e b e n d i n g / u n b e n d i n g p r o b l e m c e r t a i n s i m p l i f y i n g a s s u m p t i o n s a r e n e c e s s a r y . The m a j o r a s s u m p t i o n s a d o p t e d h e r e a r e a s f o l l o w s . 1) The d i m e n s i o n n o r m a l t o t h e n a r r o w f a c e i s n e g l e c t e d . 2) C r e e p i s n e g l e c t e d . T h us i n t h e f o r m u l a t i o n o f t h e m o d e l , t h e t w o - d i m e n s i o n a l f i n i t e - e l e m e n t p r o g r a m , E P I C - I V 1 9 , w h i c h was d e v e l o p e d by Yamada f o r p l a n e s t r e s s / p l a n e s t r a i n and a x i s y m m e t r i c p r o b l e m s , h a s b e e n u s e d w i t h some m o d i f i c a t i o n t o t a k e i n t o a c c o u n t t h e m o v i n g c o n d i t i o n o f t h e s l a b . The f i n i t e - e l e m e n t m e t h o d i s e m i n e n t l y w e l l s u i t e d f o r s o l v i n g n o n - l i n e a r a n d c o m p l e x l o a d i n g p r o b l e m s . 23 The v a l i d i t y o f t h e s e a s s u m p t i o n s a r e e x a m i n e d i n d e t a i l i n t h e s u b s e q u e n t s e c t i o n s . 3.3.1 C o m p a r i s o n o f t h e t h r e e - d i m e n s i o n a l a n d t w o - d i m e n s i o n a l m o d e l s To c h e c k t h e a d e q u a c y o f t h e t w o - d i m e n s i o n a l m o d e l , a t h r e e - d i m e n s i o n a l e l a s t i c a n a l y s i s was p e r f o r m e d and c o m p a r e d w i t h r e s u l t s f r o m t h e t w o - d i m e n s i o n a l m o d e l . A c o m p u t e r p r o g r a m , E L A S 6 5 , d e v e l o p e d by t h e C o m p u t e r S t r u c t u r a l A n a l y s i s G r o u p o f Duke U n i v e r s i t y , was u s e d f o r t h e t h r e e - d i m e n s i o n a l b e n d i n g a n a l y s i s o f t h e w i d e s l a b . A s s u m i n g s y m m e t r y , a h a l f s e c t i o n o f a s l a b was a n a l y s e d . The s l a b was c o n s i d e r e d t o be a h o l l o w box w i t h a l i n e a r t e m p e r a t u r e g r a d i e n t i n t h e t h r o u g h - t h i c k n e s s d i r e c t i o n o f t h e s h e l l . The b u l g i n g due t o f e r r o s t a t i c p r e s s u r e o f m o l t e n s t e e l was e x c l u d e d i n t h i s a n a l y s i s . A s c h e m a t i c v i e w o f t h e t h r e e - d i m e n s i o n a l f i n i t e - e l e m e n t mesh i s shown i n F i g . 3 . 5 . C a l c u l a t i o n s w e r e p e r f o r m e d f o r t h e f o l l o w i n g - c o n d i t i o n s . 1) S l a b s i z e : 2 5 0 d x l 8 0 0 W m m 2 2) S h e l l t h i c k n e s s : 90 mm 3) R o l l p i t c h : 350 mm 4) B e n d i n g r a d i u s : 10.5 m 5) M e c h a n i c a l p r o p e r t i e s : 24 ( o u t e r ) T=1045°C, E=18767 M P a , v =0.36 ( m i d d l e ) .... T=1235°C, E=15278 M P a , v =0.37 ( i n n e r ) T=1425°C, E=8163 M P a , v =0.39 Owing t o s y m m e t r y , t h e y - c o m p o n e n t o f d i s p l a c e m e n t was c o n s t r a i n e d on t h e l o n g i t u d i n a l c e n t e r p l a n e o f t h e w i d e f a c e o f t h e s l a b . I n t h e z - d i r e c t i o n , t h e nodes c o r r e s p o n d i n g t o t h e r o l l s u p p o r t i n g p o i n t s were c o n s t r a i n e d . The u p s t r e a m edge p l a n e p e r p e n d i c u l a r t o t h e c a s t i n g d i r e c t i o n was l o a d e d w i t h b e n d i n g moments b u t k e p t p l a n a r a f t e r d e f o r m a t i o n , w h i l e t h e d o w n s t r e a m edge p l a n e was l e f t f r e e . F i g . 3 . 6 shows t h e r e s u l t a n t d e f o r m a t i o n due t o b e n d i n g , a n d F i g . 3 . 7 shows t h e s t r a i n d i s t r i b u t i o n i n t h e c r o s s s e c t i o n o f t h e s l a b . Thus i t i s c l e a r t h a t t h e w i d e f a c e s h e l l s o f t h e s l a b b e n d a b o u t t h e i r i n d i v i d u a l n e u t r a l a x e s a n d t h e r e f o r e c a n be r e g a r d e d a s i n d e p e n d e n t o f t h e n a r r o w f a c e . T h i s t r e n d may be a t t r i b u t e d t o t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h e s h e l l a n d t h e s t r a n d a s p e c t r a t i o ( w / d ) f o r t h e c a s e a n a l y s e d . F i g . 3 . 8 shows a c o m p a r i s o n o f t h e r e s u l t s o f t h i s a n a l y s i s o b t a i n e d w i t h t h e t w o - d i m e n s i o n a l e l a s t i c m o d e K p l a n e s t r e s s ) o f t h e c e n t e r p l a n e o f t h e w i d e f a c e . I t i s e v i d e n t t h a t t h e r e i s g o o d a g r e e m e n t . I f p l a s t i c b e h a v i o r were t o be i n c l u d e d , t h e m i d - f a c e w o u l d a c t e v e n more i n d e p e n d e n t l y o f t h e e d g e s . T h e r e f o r e i t c a n be c o n c l u d e d t h a t t h e t w o - d i m e n s i o n a l m o d e l , a s s u m i n g p l a n e s t r e s s , i s s u f f i c i e n t f o r t h e z Fig. 3.5 Schematic Diagram of the Three-Dimensional Finite-Element Mesh for the Bending Analysis. Fig. 3.6 Predicted Distortions of the Slab by the Three-Dimensional Finite-Element Bending Analysis. ON 27 Bending S t r a i n (%) c o •H ra C Q) / 1.0 0.5* A c o •H m CQ ft E o o 0.5 -1.0 1 j ^ / ' // ^ A / / / / ' ' ' / Fig. 3.7. Predicted XX-STRAIN Distribution in the Cross Section of the Slab by the Three-Dimensional Finite-Element Bending Analysis. 28 29 b e n d i n g / u n b e n d i n g a n a l y s i s on t h e c e n t e r p l a n e o f t h e w i d e f a c e of "the s l a b . 3.3.2 E f f e c t s o f c r e e p i n c a l c u l a t i o n s o f b u l g i n g I n t h e f o l l o w i n g s e c t i o n t h e e f f e c t o f n e g l e c t i n g c r e e p i n t h e b u l g i n g a n a l y s i s on t h e a c c u r a c y o f t h e c a l c u l a t i o n s h a s b e e n e v a l u a t e d . T h i s i s a c c o m p l i s h e d by e x a m i n i n g t h e r e s u l t s o f s e v e r a l s t u d i e s on b u l g i n g i n c o n t i n u o u s l y c a s t s l a b s w h i c h h a v e i n c l u d e d c r e e p . 1 2 ' 1 5 ' 2 0 " 2 7 G r i l l a n d S c h w e r d t f e g e r 2 2 c a l c u l a t e d b u l g i n g a c c o u n t i n g f o r p r i m a r y c r e e p u s i n g a f i n i t e - e l e m e n t m o d e l a n d c o m p a r e d t h e i r r e s u l t s w i t h t h e r e s u l t s o b t a i n e d w i t h an e l a s t o - p l a s t i c m o d e l r e p o r t e d by Emi a n d S o r i m a c h i 2 0 ( s e e F i g . 3 . 9 ) . I t i s e v i d e n t f r o m t h i s c o m p a r i s o n , t h a t t h e e l a s t o - p l a s t i c m o d e l p r e d i c t s l o w e r v a l u e s o f b u l g i n g a t ' l a r g e r o l l s p a c i n g s t h a n t h e m o d e l w h i c h i n c l u d e s c r e e p b u t a t s m a l l r o l l s p a c i n g s ( l e s s t h a n 40 cm) a n d f o r s m a l l v a l u e s o f b u l g i n g ( 5 <1 mm), t h e d i f f e r e n c e b e t w e e n t h e r e s u l t s o f t h e max two m e t h o d s i s n e g l i g i b l e . To c h e c k t h e v a l i d i t y o f t h e e l a s t o - p l a s t i c m o d e l f o r t h e b u l g i n g a n a l y s i s , a c o m p a r i s o n o f m o d e l p r e d i c t i o n s t o t h e d a t a o f W u n n e n b e r g 2 7 a n d M o r i t a 2 6 was c o n s i d e r e d . However b e c a u s e Wunnenberg made m e a s u r e m e n t s w i t h l a r g e r o l l s p a c i n g s , o n l y t h e d a t a o f M o r i t a , shown i n T a b l e I I was u s e d . F o r 30 R o l l P i t c h (cm) Fig. 3.9 Comparison of Maximum Bulging Predicted by the Creep Model and Elasto-Plastic Model. c o m p a r i s o n p u r p o s e s t h e f i n i t e - e l e m e n t mesh shown i n F i g was s e l e c t e d f o r t h e e l a s t o - p l a s t i c a n a l y s i s . T a b l e I I M e a s u r e d d a t a o f b u l g i n g by M o r i t a 2 6 K i n d o f s t e e l S i - k i l l e d ( 4 0 k g / mm 2) s t e e l g r a d e S i z e 2 3 0 x 1 2 3 0 mm2 C a s t i n g s p e e d 1.1 m/min S h e l l t h i c k n e s s 55 mm S u r f a c e t e m p e r a t u r e 1 0 0 0 - 1 0 5 0 °C R o l l p i t c h 399 mm F e r r o s t a t i c h e a d 7.933 m(0. 54MPa) B u l g i n g 0 . 2-0.4 mm NODE 189 ELEMENT 320 Ferro-static Pressure i i u i i i i . j j . u n n u i 3* K t t t t K Iff filtiWliPl* 199.5 mm Temperature U65"C E E in 1050 *C Fig. 3.10 Schematic Diagram of the Two-Dimensional Finite-Element Mesh for the Bulging Analysis. 32 M e c h a n i c a l p r o p e r t i e s were o b t a i n e d f r o m t h e d a t a shown i n F i g s . 3 . 2 , a n d 3.3( s e e S e c t i o n 3.2) a s s u m i n g a u n i f o r m s t r a i n r a t e o f 1 x 1 0 " 3 s " 1 ( A p p e n d i x I ) . The r e s u l t s o f t h e p l a n e s t r e s s a n d p l a n e s t r a i n c a l c u l a t i o n s a r e shown i n F i g . 3 . 1 1 . The b u l g i n g a s s u m i n g p l a n e s t r e s s i s u s u a l l y l a r g e r t h a n t h a t b a s e d on p l a n e s t r a i n a n d t h e d i f f e r e n c e s become m o r e ' p r o n o u n c e d ' w i t h i n c r e a s e d b u l g i n g . The a g r e e m e n t b e t w e e n t h e e l a s t o - p l a s t i c a n a l y s i s and t h e m e a s u r e d b u l g i n g i s r e a s o n a b l y g o o d . The p l a n e s t r e s s c o n d i t i o n w h i c h h a s been commonly u s e d i n b u l g i n g a n a l y s i s by many a u t h o r s was a d o p t e d i n t h e p r e s e n t s t u d y . S t r i c t l y s p e a k i n g , t h e v a l i d i t y o f t h i s c o n d i t i o n w i l l d e p e n d on t h e d e g r e e o f r e s t r a i n t t h e edge e x e r t s on t h e d e f o r m a t i o n a t t h e c e n t e r o f t h e w i d e f a c e o f t h e s l a b . Thus t h e e l a s t o - p l a s t i c m o d e l h a s b e e n shown t o be r e a s o n a b l y a c c u r a t e f o r c a l c u l a t i n g b u l g i n g u n d e r c o n d i t i o n s o f s m a l l r o l l s p a c i n g s ; . i n a modern s l a b c a s t e r r o l l s p a c i n g s a r e a p p r o x i m a t e l y 30-40 cm. U n d e r t r a n s i e n t c o n d i t i o n s s u c h a s d u r i n g i n t e r r u p t i o n o f c a s t i n g t h e c r e e p m o d e l w i l l be n e c e s s a r y h o w e v e r . A n o t h e r i m p o r t a n t e f f e c t o f c r e e p on b u l g i n g i s t h a t t h e l o c a t i o n o f t h e maximum b u l g i n g s h i f t s f r o m t h e m i d p o i n t b e t w e e n r o l l s t o t h e d o w n s t r e a m a s a r e s u l t o f i n t e r a c t i o n b e t w e e n s t r a n d movement a n d c r e e p d e f o r m a t i o n . 2 2 ' 2 3 ' 2 5 ' 2 7 T h i s c a u s e s a n a s y m m e t r i c b u l g i n g d e f o r m a t i o n b u t t h e i n f l u e n c e on t h e maximum d e f l e c t i o n i s s m a l l . T h e r e f o r e t h i s e f f e c t was a l s o n e g l e c t e d i n t h e p r e s e n t s t u d y . 33 0.9 0.8 E 0.7 E 0.6 LZ 0.5 cn 0.4 CD 0.3 0.2 0.1 o Plane stress A PI ane st ra i n | data by Morita26 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 F e r r o - s t a t i c Pressure (MPa) F i g . 3.11 Comparison o f B u l g i n g S t r a i n s P r e d i c t e d by t h e P l a n e S t r e s s and P l a n e S t r a i n . F i n i t e - E l e m e n t A n a l y s i s . ^ 0.0 6 c 0.0 5 2 0.0 4 I- w 0.03 cn <= 0.0 2 D CQ 3 o.o 1h 0 o Plane stress A Plane strai n 10 20 30 40 N Number of divisions in a roll pitch F i g . 3.12 I n f l u e n c e o f t h e Mesh S i z e on B u l g i n g S t r a i n i n t h e E l a s t o - P l a s t i c F i n i t e - E l e m e n t A n a l y s i s . 34 In a f i n i t e - e l e m e n t a n a l y s i s , mesh s i z e has a s i g n i f i c a n t i n f l u e n c e on t h e r e s u l t s . The e f f e c t o f mesh s i z e i n the c a s t i n g d i r e c t i o n has been e v a l u a t e d and t h e r e s u l t s a r e shown i n F i g . 3 . 1 2 . The c o n d i t i o n s employed i n the e a r l i e r c a l c u l a t i o n s have been u t i l i z e d f o r t h i s e v a l u a t i o n . Thus twenty d i v i s i o n s i n a r o l l p i t c h can be r e g a r d e d as a s u f f i c i e n t l y f i n e mesh f o r the purpose of t h i s s t u d y s i n c e t h e r e i s l i t t l e change i n the c a l c u l a t e d r e s u l t s beyond t h i s p o i n t . Moreover t h e use of more d i v i s i o n s w i l l r e s u l t i n a p r o h i b i t i v e l y h i g h computing c o s t . Based on t h e s e a s s u m p t i o n s t h e b e n d i n g / u n b e n d i n g a n a l y s i s combined w i t h b u l g i n g was c a r r i e d out u s i n g a two- d i m e n s i o n a l ( p l a n e s t r e s s ) , e l a s t o - p l a s t i c , f i n i t e - e l e m e n t model. D e t a i l s o f the model a r e p r e s e n t e d i n subsequent s e c t i o n s . • 3.3.3 Two-dimensional e l a s t o - p l a s t i c F i n i t e Element In t h e f i n i t e - e l e m e n t method a c o n t i n u u m i s a p p r o x i m a t e d by an assemblage of e l e m e n t s which a r e i n t e r c o n n e c t e d a t a f i n i t e number of j o i n t s or n o d a l p o i n t s . By s a t i s f y i n g e q u i l i b r i u m of f o r c e s , c o m p a t i b i l i t y of d i s p l a c e m e n t s and the s t r e s s - s t r a i n law f o r t h e m a t e r i a l , i t i s p o s s i b l e t o g e n e r a t e a s e t of l i n e a r l y i n d e p e n d e n t e q u a t i o n s t h a t can be s o l v e d s i m u l t a n e o u s l y f o r t h e d i s p l a c e m e n t s a t n o d a l p o i n t s . These can then be used t o o b t a i n the s t r e s s 35 s t r a i n d i s t r i b u t i o n i n t h e assemblage of e l e m e n t s . The m a t h e m a t i c a l b a s i s of the e l a s t o - p l a s t i c f i n i t e - e l e m e n t i s d e s c r i b e d i n Appendix I I and I I I . The computer program " E P I C - I V " 1 9 has been used i n the p r e s e n t s t u d y . The main c h a r a c t e r i s t i c s of t h i s program a r e : 1) Three-noded l i n e a r t r i a n g u l a r e l e m e n t s a r e used. 2) The i t e r a t i v e method ( C o n j u g a t e G r a d i e n t method) i s adopted t o s o l v e the m a t r i x i n v e r s i o n . 3) I s o t r o p i c h a r d e n i n g of the m a t e r i a l i s assumed i n p l a s t i c i t y . 4) The i n c r e m e n t a l method ( t a n g e n t modulus method) i s adopted t o s i m u l a t e m a t e r i a l n o n - l i n e a r i t y . 5) U n l o a d i n g i s checked a t e v e r y s t a g e of t h e c a l c u l a t i o n ( Appendix I I I ) . The i t e r a t i v e m a t r i x i n v e r s i o n p r o c e d u r e g r e a t l y r e d u c e s the computing time f o r a n o n - l i n e a r p r o b l e m . The d i s a d v a n t a g e of t h i s t e c h n i q u e i s t h a t when the m a t r i x t o be i n v e r t e d a p p r o a c h e s s i n g u l a r i t y ( i . e . p l a s t i c i n s t a b i l i t y ) t he convergence d e t e r i o r a t e s . In o r d e r t o use t h i s program t h e f o l l o w i n g m o d i f i c a t i o n s were made. To f a c i l i t a t e t h e semi-dynamic s i m u l a t i o n , the main r o u t i n e which c o n t r o l s t h e s u b r o u t i n e s was changed. By t h i s m o d i f i c a t i o n a s h i f t i n t h e boundary c o n d i t i o n ( r o l l s u p p o r t i n g p o i n t s ) was made p o s s i b l e . S e c o n d l y , sub 36 programs f o r p l o t t i n g the r e s u l t s were f o r m u l a t e d as a p o s t d a t a p r o c e s s o r . The f o l l o w i n g p l o t s a r e a v a i l a b l e : f i n i t e - element mesh, d e f o r m a t i o n s , p r i n c i p a l s t r a i n v e c t o r s and c o n t o u r maps of s t r e s s e s and s t r a i n s . To check t h e a c c u r a c y of t h e p r ogram,the s t r e s s / s t r a i n d i s t r i b u t i o n i n a t h i c k - w a l l e d c y l i n d e r under i n t e r n a l p r e s s u r e was computed and compared w i t h t h e a n a l y t i c a l s o l u t i o n by H i l l ( A p p e n d i x I V ) . 3.3.4 Boundary c o n d i t i o n s The l o n g i t u d i n a l c e n t e r p l a n e of t h e wide f a c e of a s l a b has been m o d e l l e d . F i g . 3 . 1 3 shows a s c h e m a t i c v i e w of the p l a n e o f i n t e r e s t . Three and one h a l f r o l l p i t c h e s i n the s t r a i g h t e n i n g zone were m o d e l l e d and t h e upper and l o w e r s h e l l s o f t h i s domain were a n a l y s e d s e p a r a t e l y u s i n g t h e f o l l o w i n g boundary c o n d i t i o n s . 1 ) The x- and y-ccmponents of the d i s p l a c e m e n t s of the nodes on the downstream edge were s e t e q u a l t o the v a l u e s of g e o m e t r i c a l d i s p l a c e m e n t s b a s e d on s i m p l e beam b e n d i n g t h e o r y . 2) The y-component o f d i s p l a c e m e n t o f r o l l s u p p o r t i n g p o i n t s were c o n s t r a i n e d . The y-component of nodes a l o n g AB and CD were a l s o c o n s t r a i n e d ( F i g . 3 . 1 3 ) , s i n c e s t r a i g h t e n i n g i s assumed t o be c o m p l e t e d w i t h i n the f i r s t r o l l p i t c h downstream f r o m t h e b e n d i n g F i g . 3.13 S c h e m a t i c Diagram o f t h e Boundary C o n d i t i o n s A d o p t e d i n t h e T w o - D i m e n s i o n a l F i n i t e - E l e m e n t B e n d i n g A n a l y s i s . 38 p o i n t . 3 1 3) The r o l l f r i c t i o n f o r c e s c a u s e d by b e n d i n g d e f o r m a t i o n were u n i f o r m l y d i s t r i b u t e d b e t w e e n a d j a c e n t r o l l s . ( C o e f f i c i e n t o f r o l l f r i c t i o n was a s s u m e d t o be 0.45) 4) The u p s t r e a m edge o f t h e s h e l l was c o n s t r a i n e d t h r o u g h t h e f o r c e b o u n d a r y c o n d i t i o n ; t h i s i s e q u i v a l e n t t o t h e c o n s t r a i n t f o r c e f r o m t h e r e m a i n i n g doma i n . 5) The r o l l p o i n t s w e re s h i f t e d o n c e i n t h e b e n d i n g a n a l y s i s a n d t h e a b o v e b o u n d a r y c o n d i t i o n s were r e a p p l i e d . 3.3.4.1 R o l l f r i c t i o n f o r c e The c r o s s s e c t i o n p e r p e n d i c u l a r t o t h e c a s t i n g d i r e c t i o n d o e s n o t r e m a i n p l a n a r a f t e r t h e b e n d i n g d e f o r m a t i o n ( see F i g . 3 . 6 o f S e c t i o n 3 . 3 . 1 ) . I f t h e s e c t i o n u n d e r c o n s i d e r a t i o n were f r e e a t t h e e n d s , t h e c e n t e r p l a n e o f t h e w i d e f a c e o f t h e u p p e r s h e l l w o u l d move d o w n s t r e a m r e l a t i v e t o t h e n a r r o w f a c e d u r i n g u n b e n d i n g , w h i l e t h e c e n t e r p l a n e o f t h e l o w e r s h e l l w o u l d move u p s t r e a m . T h i s t e n d e n c y o f t h e c e n t e r p l a n e t o move r e l a t i v e t o t h e n a r r o w f a c e i s o p p o s e d by f r i c t i o n a l f o r c e s b e t w e e n t h e r o l l s a n d t h e s t r a n d s u r f a c e , s e e F i g . 3 . 1 3 . The r o l l f r i c t i o n f o r c e h a s been e s t i m a t e d a c c o r d i n g t o t h e f o l l o w i n g s t e p s : 39 1) C a l c u l a t e t h e a v e r a g e movement o f t h e d o w n s t r e a m edge /TT . 2) Assume t h e number o f r o l l s , N , n e c e s s a r y t o a b s o r b t h e a b o v e d i s p l a c e m e n t ,~u x. N 3) C a l c u l a t e t h e c u m u l a t i v e r o l l f r i c t i o n f o r c e s , %F±. i = l F. = V X R i ( 1 2 ) where p : f e r r o s t a t i c p r e s s u r e i 1 : r o l l p i t c h R i y : f r i c t i o n a l c o e f f i c i e n t (=0.45 ) 4) C o n v e r t E t o s t r e s s ° i a t e a c h r o l l p o i n t . c I =( £ F i / S h e l l ) x 4 ( 1 3 ) ( s t r e s s CTi i s t h e t o t a l o f f o u r l a y e r s o f d i f f e r e n t m a t e r i a l s . ) 5) D e c i d e s t r a i n c o r r e s p o n d i n g t o t h e s t r e s s a. i f r o m t h e s t r e s s - s t r a i n c u r v e , 6) C a l c u l a t e e l o n g a t i o n A l f r o m s t r a i n A 1 i = XR1 ' E i * 1 0 2 ( 1 4 ) 7). R e p e a t p r o c e d u r e s f r o m 2) u n t i l f i n a l c o n v e r g e n c e i s a c h i e v e d . \ N 1 h = \ ( 1 5 ) i = l 1 x 40 Appendix V presents an example of t h i s c a l c u l a t i o n of Case 1( see Table III of Sect ion 4) , where the value of u x i s 8 .86 mm and N i s 11. The value of the c o e f f i c i e n t of r o l l f r i c t i o n s t rong ly a f f ec t s the r e s u l t a n t bending s t r a i n as shown in F i g . 3 . 1 4 . the c a l c u l a t i o n c o n d i t i o n s for which are the same as i n Case 2( see Table III of Sect ion 4) . Unfor tunate ly there i s no measured data a v a i l a b l e for the c o e f f i c i e n t of r o l l f r i c t i o n for the continuous cas t ing of s labs . The value of 0.33 has been adopted e m p i r i c a l l y for the design of d r i v i n g r o l l s ; however t h i s value appears to be an underestimate to prov ide a margin of safety in the des ign. In the present a n a l y s i s , t h e r e f o r e , the data summarized by S c h e y 3 2 for hot r o l l i n g was used, see F i g . 3 . 1 5 . The f r i c t i o n i s seen to decrease with i n c r e a s i n g temperature; and therefore the iron oxide f i l m th i cknes s i s reported to be one of the main v a r i a b l e s which a f f e c t s f r i c t i o n . The value of 0.45( see F i g . 3 . 1 5 , Pavlov in a i r ) corresponding to the surface temperature of 9 5 0 ° C was adopted for the present a n a l y s i s , s ince in the one-point bending bow- type cas ters the surface temperature of the s t r a n d ranges from about 9 0 0 ° C to 1 0 0 0 ° C at the bending p o i n t . 1 2 3 .3 .4 .2 S h i f t of the boundary c o n d i t i o n To take into account the e f f e c t s of a moving 41 B.i P i i x i -1 0 1 Roll No..at s t ra ightening zone Fig. 3.14 Influence of Coefficient of Roll Friction on the Resultant Bending Strain. 42 Temperature ( ° C ) Fig. 3.15 Coefficient of Roll Friction of Hot Rolling as a Function 3 ? Temperature.J 43 Fig. 3.16 Predicted Bending Strain with the One-Step Bending Model. 44 Roll No. at straightening zone -10" inner surface C D outer surface c c CD -1.5 -CD 5 • CASE 1 -2.0.1 1 ' L. Fig. 3.17 Predicted Bending and Bulging Strain with the One-Step Bending Model. 45 s t r a n d , t h e r o l l p o i n t s w e r e s h i f t e d d u r i n g b e n d i n g . F o r a c o m p l e t e d y n a m i c m o d e l l i n g i t i s n e c e s s a r y t o s h i f t r o l l p o i n t s b y s m a l l s t e p s ; h o w e v e r t h i s r e s u l t s i n a p r o h i b i t i v e l y h i g h c o m p u t i n g c o s t . I n t h e p r e s e n t a n a l y s i s , t h e r o l l p o i n t s h a v e b e e n s h i f t e d o n l y o n c e d u r i n g b e n d i n g a n d h e n c e t h e s t r a n d was b e n t i n t w o - s t e p s a r o u n d t h e two b e n d i n g p o i n t s ( s e e F i g . 4 . 9 o f S e c t i o n 4 ) . A s i g n i f i c a n t d i f f e r e n c e was o b s e r v e d i n t h e r e s u l t s o f t w o - s t e p b e n d i n g c o m p a r e d t o t h o s e o f s i n g l e - s t e p b e n d i n g , e s p e c i a l l y i n t h e c a s e o f t h e l o w e r s h e l l . Compare F i g s . 3 . 1 6 , 3 . 1 7 w i t h F i g s . 5 . 1 2 , 5 . 2 0 ( C h a p t e r 5) a n d n o t e t h a t t h e b e n d i n g s t r a i n d i s t r i b u t i o n i n t h e i n n e r a n d o u t e r s u r f a c e a r e s m o o t h e r f o r t h e c a s e w h e r e a t w o - s t e p b e n d i n g c a l c u l a t i o n p r o c e d u r e was e m p l o y e d . I n t h e c a s e o f t h e o n e - s t e p b e n d i n g c a l c u l a t i o n , p e a k s t r a i n s a p p e a r i n t h e i n n e r s u r f a c e w h i c h w e r e m a g n i f i e d a s a r e s u l t o f i n t e r a c t i o n b e t w e e n b e n d i n g a n d b u l g i n g ,see F i g s . 3 . 1 6 , 3 . 1 7 . I t i s b e l i e v e d t h e s e peak s t r a i n s c a u s e a s i g n i f i c a n t e r r o r i n t h e r e s u l t s o f o n e - s t e p b e n d i n g . 3.3.5 C a l c u l a t i o n f l o w F i g s . 3 . 1 8 and 3.19 shows t h e f l o w c h a r t f o r t h e c a l c u l a t i o n . " B e n d i n g " a n d " b e n d i n g p l u s b u l g i n g " w e r e c a l c u l a t e d s e p a r a t e l y . E a c h c a l c u l a t i o n c o n s i s t s o f two s t e p s , i . e . f i r s t - s t e p b e n d i n g a n d s e c o n d - s t e p b e n d i n g , s e e F i g . 3 . 1 8 . A f t e r t h e f i r s t - s t e p b e n d i n g , r o l l p o i n t s were s h i f t e d by a h a l f p i t c h t o s e t new b o u n d a r y a n d l o a d i n g c o n d i t i o n s f o r 46 the s e c o n d - s t e p b e n d i n g a n a l y s i s . The nodes of t h e former r o l l p o i n t s were u n l o a d e d t o a l l o w f o r a s p r i n g back. The nodes of the new r o l l p o i n t s were l o a d e d w i t h the d i s p l a c e m e n t s , U Y /, t o push t h e s t r a n d back t o s i m u l a t e r o l l c o n s t r a i n t a t t h e new r o l l p o i n t s . In the c a l c u l a t i o n of "bending p l u s b u l g i n g " , f e r r o s t a t i c p r e s s u r e was l o a d e d i n the s e c o n d - s t e p b e n d i n g s t a g e . 47 c READ INPUT DATA GENERATE MESH TOPOLOGY NODE COORDINATES c PRINT INPUT DATA 'ELASTO-PLASTIC ROUTINE" c PRINT RESULTS 3 MOVE ROLLER POINTS BY A HALF PITCH SET NEW BOUNDARY & LOADING CONDITIONS I | APPLY FERRO-STATIC J J PRESSURE j •ELASTO-PLASTIC ROUTINE" c PRINT/PLOT RESULTS 3 Fig. 3.18 Flow Chart for the Calculation of the Bending and Bulging Strain. 48 1 S T A R T E L E M E N T C O U N T N=1 ,2 N E C O M P U T E E L E M E N T S T I F F N E S S M A T R I X A S S E M B L E M A S T E R S T I F F N E S S M A T R I X Y E S S O L V E E O U A T I O N S B Y C . G . M E T H O D V C A L C U L A T E O F S T R E S S I N C R E M E N T S A N D S T R A I N C A L C U L A T E T H E M I N I M U M L O A D R A T I O ( R M | ( ( ) F O R E L E M E N T S D O A N Y E L E M E N T S S^. R E C O V E R E L A S T I C A L L Y 7 ^ Y E S MIX MIM A D D T H E I N C R E M E N T S T O T H E P R E V I O U S V A L U E S N O 19 Flow Chart for the Calculation of the Bending and Bulging Strain. ( "Elasto-Plastic Routine" > 49 Chapter 4 CALCULATION CONDITIONS C a l c u l a t i o n s have been p e r f o r m e d o n l y f o r o n e - p o i n t b e n d i n g bow-type c a s t e r s i n an a t t e m p t t o o b t a i n a f u n d a m e n t a l u n d e r s t a n d i n g of bending/unbending of c o n t i n u o u s l y c a s t s l a b s . M u l t i b e n d i n g bow-type c a s t e r s have not been c o n s i d e r e d s i n c e a n a l y s i s of such machines s h o u l d a l l o w f o r s t r e s s r e l a x a t i o n due t o c r e e p t h u s making c r e e p a n a l y s i s mandatory. The main p a r a m e t e r s t h a t were i n v e s t i g a t e d w i t h the computer model a r e as f o l l o w s ; 1) Machine R a d i u s , R 2 ) R o l l P i t c h , 1 R 3) C a s t i n g Speed, V 4) S h e l l T h i c k n e s s , s 5 ) S u r f a c e Temperature, T Q 6) F e r r o s t a t i c P r e s s u r e , p 50 However t h e t h r e e p a r a m e t e r s - f e r r o s t a t i c p r e s s u r e , s h e l l t h i c k n e s s a n d s u r f a c e t e m p e r a t u r e a r e n o t i n d e p e n d e n t v a r i a b l e s ; t h e l a t t e r two a r e d e p e n d e n t upon m a c h i n e r a d i u s and c a s t i n g s p e e d w h i l s t f e r r o s t a t i c p r e s s u r e i s o n l y d e p e n d e n t upon m a c h i n e r a d i u s . The s h e l l t h i c k n e s s a n d s u r f a c e t e m p e r a t u r e h a v e been o b t a i n e d f r o m t h e p l a n t d a t a a t O i t a w o r k s ( N S C ) * 6 , where t h e s u r f a c e t e m p e r a t u r e h a s been s m o o t h e n e d f o r s i m p l i f i c a t i o n . A p l o t o f s h e l l t h i c k n e s s and m i d - f a c e t e m p e r a t u r e a g a i n s t t i m e (= a x i a l d i s t a n c e / c a s t i n g s p e e d ) i s shown i n F i g . 4 . 1 . The d e s i g n a n d o p e r a t i n g c o n d i t i o n s o f t h e s l a b c a s t e r a t O i t a w o r k s o f N i p p o n S t e e l C o r p o r a t i o n , N S C 3 5 , were c h o s e n a s a b a s e c a s e f o r t h e c a l c u l a t i o n t o e n a b l e a c o m p a r i s o n t o be made b e t w e e n m o d e l p r e d i c t i o n s a n d p l a n t d a t a on i n t e r n a l c r a c k s r e s u l t i n g f r o m t h e u n b e n d i n g . The m a c h i n e s p e c i f i c a t i o n s o f O i t a No.4 c a s t e r a r e a s f o l l o w s ( s e e a l s o F i g . 4 . 2 ) ; I ) m a c h i n e r a d i u s : 10.5 m I I ) r o l l p i t c h : 47 1 mm I I I ) s l a b s i z e : 2 5 0 d x 1300 - l 9 0 0 W m m I V ) c h e m i c a l c o m p o s i t i o n o f s l a b ( A l - S i - k i l l e d ( 4 0 k g / m m 2 ) s t e e l g r a d e ) : 0 . 1 5 / 0 . 1 9 % C , 0 . l O / 0 . 3 0 % S i , 0 . 7 0 / 0 . 9 0 % M n , 0 . 0 2 5 % > P , O . O l 5 % ^ S , O . O l / 0 . 0 4 % T , A l V) t h r e s h o l d c a s t i n g s p e e d f o r b e n d i n g i n t e r n a l c r a c k s : 1.1 - 1.2 m/min. o 1 5 0 0 ""'HOO 0 2 4 6 8 10 12 14 16 18 • 20 22 Time below meniscus ( min.) g. 4.1 Surface Temperature and Shell Thickness in the Continuous Casting of Slab. Fig. 4.2 Roll Profile of the 10.5 1. B. Ferro-static Pressure (M a O 3 Roll Pi tch ( mm ) Radius Caster, Table III Calculation conditions for unbending of continuously cast slabs. Machine Radius R m Roll Pitch HR Casting Speed v m/min Shell Thickness S mm Surface Temp. To 'C Ferrostatic Pressure P MPa CASE 1. U,L 10.5 471 1.6 83 930 0.74 2. U 10.5 471 1.2 * 97 900 0.74 3. U,L 10.5 471 1.0 106 900 0.74 4. U 10.5 471 1.2 97 990 0.74 5. U . 10.5 471 1.2 97 850 0.74 6. U 10.5 400 1.2 97 900 0.74 7. U 10.5 540 1.2 97 900 0.74 8. U 8.0 471 0.9 97 900 0.56 9. U 8.0 471 1.2 83 900 0.56 10. U 13.0 471- 1.47 97 900 0.91 TJ : upper shell , L : lower shell * : threshold casting speed for internal cracks to Ferro-static Pressure (MPa) LT ID ir in d d cr LT LT <3 a cr LT) tTTTTTTTTTTJ CT LT a in E=L\ Roll Pitch (mm) Roll Profile of the 8.0 m Radius Caster. Fig. 4.4 Roll Profile of the t B. P Ferro-static Pressure (MPa) Roll Pitch (mm) Radius Caster. 56 The conditions which were investigated in the present study are given in Table III. The slab thickness was kept constant at 250 mm for a l l cases. In Cases 1,2 and 3, the casting speed was varied to evaluate the c r i t i c a l strain necessary for generation of internal cracks. In Cases 4 and 5, the surface temperature was a r t i f i c i a l l y changed using the same conditions as in Case 2; and in Cases 6 and 7, the r o l l pitch was changed. In Cases 8,9 and 10, machine radii of 8 m and 13 m were studied; the r o l l configuration for these hypothetical machines is shown in Figs.4.3 and 4.4. The temperature distribution through the shell thickness has been assumed to be linear with the inner shell surface at the solidus temperature of A l - S i - k i l l e d (40kg/mm2)steel grade - 1487°C 3 3. Based on temperature distributions calculated separately with a heat-flow model , this assumption is very reasonable. The temperature and shell thickness were assumed to be uniform in the casting direction over the three-and-one-half r o l l pitches being considered by the model. This again is a very reasonable approximation. The mechanical properties for the above temperature distribution have been calculated from the mechanical property data shown in Figs.3.2 and 3.3( see Section 3.2.2) and different properties were assigned,to each of the four layers into which the shell was divided. To determine the strain hardening exponent,n,the strain rate in each of the layers was 57 c a l c u l a t e d by c o n s i d e r i n g t h e n e u t r a l p l a n e o f b e n d i n g t o be h a l f w a y t h r o u g h t h e s l a b t h i c k n e s s . S t r i c t l y s p e a k i n g t h e r e a l s t r a i n r a t e s must be u s e d t o d e t e r m i n e n, h o w e v e r t h i s a p p r o x i m a t i o n i s s u f f i c i e n t s i n c e t h e d a t a o f n i t s e l f h a s some s c a t t e r a s shown i n F i g . 3 . 3 . F i g s . 4 . 5 t o 4.8 show t h e s t r e s s - s t r a i n c u r v e s c a l c u l a t e d by t h i s p r o c e d u r e . The f i n i t e - e l e m e n t mesh f o r t h i s c a l c u l a t i o n i s shown i n F i g . 4 . 9 . The t o t a l number o f n o d e s a m o u n t e d t o 536 a n d t h e number o f e l e m e n t s e q u a l l e d 924. _ 50 0.1 0.2 0.3 OA 0.5 0.6 S t r a i n ( %) 0.7 0.8 0.9 1.0 Fig. 4.5 Assumed Stress-Strain Curves for the Slab in Case 1. oo 60 50 £ 40 E Z 30 N o . T ( ° C ) 6 ( V S ) 1 1413 10~5 tT=6.0£a11 2 1266 -4 10 (T=22.5£alA 3 1120 10"A 0*=62.7£a21 4 974 10' A CJzlOlOE0-23 0.3 0.4 0.5 0.6 S t r a i n ( % ) 0.7 0.8 0.9 1.0 F i g . 4.6 Assumed S t r e s s - S t r a i n C u r v e s f o r t h e S l a b i n Case 2,3,6,7,8,9 and 10. No. T (°C) 6 ( >s) 50 1 1424 10~ 5 0" = 5.1£ Q 1 1 2 1300 10~ 4 CJ=17.6E a i 3 40 - 3 1176 1 0 " 4 CJ =48.06 a 2 4 1052 10" A 0"=85.2E Q 2 3 S t r a i n ( % ) Fig. 4.7 Assumed Stress-Strain Curves for the Slab in Case 4. o 0.2 0.3 0.4 0.5 0.6 S t r a i n ( % ) 0.7 0.8 0.9 1.0 F i 8. 4.8 Assumed Stress-Strain Curves for the Slab in Case 5, N O D E 536 \ /' 1 STAGE UNBENDING E L E M E N T 924 STAGE UNBENDING Ferro-stat ic pressure —> X Fig. 4.9. Schematic Diagram of the Two-Dimensional Finite-Element Mesh for the Bending and Bulging Analysis. to 63 C h a p t e r 5 MODEL PREDICTIONS AND DISCUSSION 5.1 R e s u l t s o f c a l c u l a t i o n s F i g s . 5 . 1 t o 5.6 show t h e r e s u l t s o f b e n d i n g a nd b u l g i n g a n a l y s i s f o r Case 1 , p r e s e n t e d i n t e r m s o f t h e c o m p u t e r p l o t s o f d e f o r m a t i o n , X X - s t r a i n c o n t o u r s , X Y - s t r a i n c o n t o u r s , e f f e c t i v e s t r e s s c o n t o u r s a n d p r i n c i p a l s t r a i n v e c t o r s . As s e e n f r o m F i g s . 5 . 5 a n d 5.6, t h e d i r e c t i o n s o f p r i n c i p a l s t r a i n v e c t o r s a r e t h e same a s t h o s e o f XX- a n d Y Y - s t r a i n . T h u s , t h e c x c o m p o n e n t o f s t r a i n i s a p r i n c i p a l s t r a i n a n d w i l l be d i s c u s s e d h e r e a f t e r w i t h r e f e r e n c e t o i n t e r n a l c r a c k s . H i g h p e a k s o f t e n s i l e s t r a i n , e x ( 0 . 5 5 - 0.65%) o c c u r a t t h e i n n e r s u r f a c e o f t h e u p p e r s h e l l b e n e a t h t h e r o l l s u p p o r t p o i n t s i n F i g . 5 . 5 , w h i l e a t t h e i n n e r s u r f a c e o f t h e l o w e r s h e l l t h e v a l u e o f t h e peak s t r a i n , E i s r a t h e r s m a l l ( a b o u t 0 . 1 % ) , a s shown i n F i g . 5 . 6 . T h i s i m p l i e s t h a t i n t e r n a l c r a c k s c o u l d be e x p e c t e d t o a p p e a r p r e f e r e n t i a l l y on t h e u p p e r s h e l l i n t h e s t r a i g h t e n i n g z o n e . The d e f o r m a t i o n o f t h e u p p e r and l o w e r s h e l l s i s n o t Fig. 5.1 Predicted Distortion Due to Bending and Bulging in Case 1. iB.P. XX STRAIN 1 E S -0.3944 E-•04 2 Z S 0.1036 E-•02 3 0.2112 E--02 n 0.3188 E--02 5 0.4264 E--02 6 «•>• 0.5340 E--02 7 0.6416 E--02 8 = 0.7491 E--02 9 S 3 0.8567 E--02 10 CS 0.9643 E--02 XX STRAIN 1 _ -0.1273 E- 01 2 as -0.1120 E-•01 3 = -0.9666 E-•02 4 -0.8132 E--02 5 = -0.6598 E--02 6 as -0.5064 E--02 7 -0.3530 E--02 8 « -0.1996 E--02 9 ea -0.4625 E--03 10 et 0.1071 E--02 Fig. 5.2 Predicted XX-STRAIN Contours Due to Bending and Bulging in Case 1. ON XY STRAIN 1 = - 0 . 1 0 0 9 E - - 0 2 2 - 0 . 6 5 5 2 E - - 0 3 3 - - 0 . 3 0 1 6 E - - 0 3 4 - 0 . 5 1 9 5 E - - 0 4 5 S3 0 . 4 0 5 5 E - - 0 3 6 - 0 . 7 5 9 1 E - - 0 3 7 = 0 . 1 1 1 3 E - - 0 2 8 0 . 1 4 6 6 E - - 0 2 9 = 0 . 1 8 2 0 E - - 0 2 1 0 S3 0 . 2 1 7 4 E - - 0 2 I 2 3 4 5 6 7 8 9 1 0 XY STRAIN - 0 . 3 4 7 7 - 0 . 2 9 4 0 - 0 . 2 4 0 2 - 0 . 1 8 6 5 - 0 . 1 3 2 7 - 0 . 7 8 9 5 - 0 . 2 5 1 9 0 . 2 8 5 6 0 . 8 2 3 1 0 . 1 3 6 1 E - 0 2 E - 0 2 E - 0 2 E - 0 2 E - 0 2 E - 0 3 E - 0 3 E - 0 3 E - 0 3 E - 0 2 Fig. 5.3 Predicted XY-STRAIN Contours Due to Bending and Bulging in Case 1. i \ i L T i EFFECTIVE STRESS (MPa) 1 2 3 4 5 6 7 8 9 10 0.4806 0.7817 0.1082 0.1383 0.1684 0.1986 0.2287 0.2588 0.2889 0.3190 E+01 E+01 E+02 E+02 E+02 E+02 E+02 E+02 E+02 E+02 I ^ EFFECTIVE STRESS (MPa) 1 0 .5478 E+01 2 = 0 .8607 E+01 3 = 0 .1174 E+02 4 = 0 .1486 E+02 5 = 0 .1799 E+02 6 = 0 .2111 E+02 7 = 0 .2425 E+02 8 = 0 .2738 E+02 9 = 0 .3050 E+02 10 = 0 .3364 E+02 Fig. 5.4 Predicted EFFECTIVE STRESS Contours Due to Bending and Bulging in Case 1. ON 1.0 % i J3. P/ Compression Tens ion r l l s l i i i , . — — i - 4 —| — |— +- -.. — | — .— -> — — _ — — - 1 1 *— 1— —i- —• — . — -= : ^ 1. - I • — i — * _ _ L 'J J _ I [ | — -4— i i , K  U I" -r "I - I " F - - T " - - ~ ^ ~* 1 J J _ 1 _ L 1 J _ I _ 1_ _L v 1 I I I .1 I Fig. 5.5 Predicted Principal Strain Vectors Due to Bending and Bulging in Case 1.(Upper Shell) oo 1 . 0 % ^ — ^ Compression Tension i i r 1 R U" Iv- l**- yi 1 . 1 _ L 1 j I . I I I I f I I I I Lv- J**- I'- V - -V- 4̂ -̂ -ty - r - + "I I H " -fV- -vV- - r V 7V- -/V -Ar -A*- -!v- -V" r _ r _ i _ i_ .t _, _ , _ . h - 4 - -V >- -V -V- - ] 4 / — — f V - - f V — f V — j V - -rV -V 1 — i - -I — i - r i ~ — 1 — 1— -«—i - - t — i — ,— — — — — — — — 1 - 4 - Fig.5 .6 Predicted Principal Strain Vectors Due to Bending and Bulging in Case 1.(Lower Shell) ON Tnblc IV Strains at so l i d i f i c a t i o n front on the center plana normal to the wide face. Bulg ing Ber iding Bending and Bu lg ing inner outer e * X surface sur face e * X X * * E z E xy e" % P CASE 1. U 0.08 0.17 0.84 0.55/0.65 -0.26/-0.28 -0.27/-0.33 0.043/0.16 3.1/3.4 0.5/0.67 L 0.08 -0.21 -0.82 0.1 -0.0075 -0.085 0.09 3.8 0.56 2. U 0.062 0.1 0.85 0.25/0.3 * -0.13/-0.18 -0.12/-0.11 0.043/0.06 3.2/3.7 0.23/0.29 3. U 0.045 0 0.84 0.15/0.2 -0.078/-0.12 -0.08/-0.1 0.033/0.016 3.2/3.3 0.14/0.2 ' L 0.045 - 0 . 0 2 -0.85 0.1/0.13 -0.037/-0.09 -0.086/-0.04 0.065/0.055 3.5/3.1 0.28/0.16 4. U 0.072 0.15 0.91 0.38/0.4 -0.19/-0.15 -0.13/-0.17 0.0058/0.093 2.8/2.9 0.3/0.32 5. U 0.055 0.03 0.79 0.15/0.23 -0.077/-0.15 -0.076/-0.09 0.011/-0.001 2.4/4.0 0.14/0.34 6. U 0.034 0.09 0.84 0.15/0.20 -0.06/-0.12 -0.08/-0.09 0/0.012 3.5/3.3 0.17/0.18 7. U 0.089 0.1 0.87 0.3/0.4 -0.11/-0.25 -0.12/-0.17 0.055/0.058 3.4/3.6 0.27/0.42 8. U 0.039 0.09 1.1 0.15/0.2 -0.086/-0.098 -0.077/-0.10 0.0075/0.004 2.9/3.6 0.22/0.17 9. U 0.063 0.19 1.05 0.35/0.4 -0.19/-0.15 -0.17/-0.084 0.007/-0.04 3,3/3.6 0.33/0.27 10. u 0.082 0.11 0.73 0.3/0.33 -0.11/-0.16 -0.13/-0.12 0.056/0.0057 3.2/3.5 0.24/0.26 U : upper s h e l l , L : lower s h e l l •» c r i t i c a l s t r a i n f o r i n t e r n a l cracks 71 s y m m e t r i c a l , a s shown i n F i g . 5 . 1 . Maximum b u l g i n g d e f l e c t i o n o f t h e u p p e r s h e l l i s u s u a l l y l a r g e r t h a n t h a t o f t h e l o w e r s h e l l , f o r e x a m p l e t h e maximum b u l g i n g o f t h e u p p e r a n d l o w e r s h e l l s i s 0.87 mm a n d 0.39 mm r e s p e c t i v e l y b e t w e e n N o . l a n d No.2 r o l l s . As s e e n f r o m t h e E c o n t o u r s , F i g . 5 . 2 , t h e s p a c i n g o f e a c h c o n t o u r i s u n i f o r m t h r o u g h t h e s h e l l t h i c k n e s s , i n d i c a t i n g t h a t t h e l o n g i t u d i n a l s t r a i n d i s t r i b u t i o n £ i s l i n e a r t h r o u g h X t h e s h e l l t h i c k n e s s . T h u s , t h e b e h a v i o r o f e a c h s h e l l i s s i m i l a r t o t h a t o f a s i m p l e beam r a t h e r t h a n o f a t w o - d i m e n s i o n a l c o n t i n u u m . The s h e a r s t r a i n e i s f a i r l y s m a l l , c l o s e t o one f o u r t h o f t h e c o m p o n e n t , s e e F i g . 5 . 3 . T a b l e I V l i s t s a l l t h e c o m p o n e n t s o f s t r a i n a t t h e s o l i d i f i c a t i o n f r o n t f o r t h e o t h e r c a s e s . The r e m a i n i n g r e s u l t s o f C a s e 2 t o C a s e 10 a r e p r e s e n t e d i n A p p e n d i x V I . 5.1.1 C o m p a r i s o n o f m o d e l p r e d i c t i o n w i t h p l a n t d a t a F i g . 5 . 7 shows t h e r e l a t i o n b e t w e e n i n t e r n a l c r a c k s a n d c a s t i n g s p e e d a t O i t a NO.4 c a s t e r , N S C . 3 " ' 3 5 H e r e , t h e r a t i n g o f i n t e r n a l c r a c k s i s d e f i n e d a s c r a c k l e n g t h d i v i d e d by c r a c k s p a c i n g . T h u s t h e t h r e s h o l d c a s t i n g s p e e d f o r b e n d i n g r e l a t e d i n t e r n a l c r a c k s i s s e e n t o be 1.1 - 1.2 m/min f o r t h e c a s e o f o r d i n a r y c a s t i n g i . e . w i t h o u t c o m p r e s s i o n . F i g s . 5 . 8 t o 5 . 1 0 0 6 show e x a m p l e s o f s u l f u r p r i n t s o f a l o n g i t u d i n a l s e c t i o n t h r o u g h s l a b s w i t h i n t e r n a l c r a c k r a t i n g s o f 0.2,0.5 a n d 1.0 72 r e s p e c t i v e l y . The i n t e r n a l c r a c k s g e n e r a l l y a p p e a r a s d a r k s e g r e g a t i o n l i n e s between, t h e p r i m a r y arms o f d e n d r i t e s . As i s a p p a r e n t f r o m t h e s e p r i n t s , i n t e r n a l c r a c k s t e n d t o a p p e a r on t h e u p p e r s h e l l w i t h an i n c r e a s e i n c a s t i n g s p e e d . E 1.0 r  0.8 1.0 1.2 1.4 16 1.8 Cuiing «peed (tn/min) Q Ordinary casting, Q Compression casting Fig.5.7 Relation between Internal Cracks and Casting Speed at Oita NO.4 caster, C A l - S i - k i l l e d (AOkg/mm ) steel grade) F i g s . 5 . 1 1 t o 5.15 show t h e m o d e l p r e d i c t i o n s f o r c a s t i n g s p e e d s o f 1.6,1.2 a n d 1.0 m/min. The l e v e l o f s t r a i n e x a t t h e i n n e r s u r f a c e o f t h e u p p e r s h e l l i n c r e a s e s w i t h an i n c r e a s e i n c a s t i n g s p e e d , w h i l e e a t t h e i n n e r s u r f a c e o f t h e l o w e r s h e l l r e m a i n s l o w . T h u s t h e c r i t i c a l s t r a i n f o r i n t e r n a l c r a c k s , w h i c h i s r e a c h e d a t a c a s t i n g s p e e d o f 1.2m/min, i s 0.25 t o 0.30% a c c o r d i n g t o t h e p r e s e n t a n a l y s i s . T h i s v a l u e o f c r i t i c a l s t r a i n i s r e s o n a b l e i n c o m p a r i s o n t o t h e e x p e r i m e n t a l v a l u e s r e p o r t e d i n t h e l i t e r a t u r e ( r e f e r t o S e c t i o n 73 2l)12»15t36-39 5.1.2 B u l g i n g s t r a i n F i g s . 5 . 1 6 t o 5 . 1 8 show t h e r e s u l t s of the b u l g i n g a n a l y s i s of Case 1 ( l o w e r s h e l l ) . From F i g . 5 . 1 7 , the peak t e n s i l e s t r a i n , e x , a p p e a r s p e r i o d i c a l l y a l o n g t h e i n n e r s u r f a c e of t he s h e l l . The magnitude of the shear s t r a i n , e x y i s about one h a l f of t h e E x component i n the p r e s e n t case and hence i s f a i r l y i m p o r t a n t i n t h e b u l g i n g a n a l y s i s . A s i m i l a r f i n d i n g has been r e p o r t e d by M a t s u m i y a . 2 5 T a b l e V shows the maximum b u l g i n g d e f l e c t i o n between No.1 and No.2 r o l l s f o r the c a s e s of b u l g i n g a l o n e and b u l g i n g i n c o m b i n a t i o n w i t h b e n d i n g . The t o t a l d e f l e c t i o n due t o t h e c o m b i n a t i o n of bend i n g and b u l g i n g i s u s u a l l y l a r g e r than t h a t due t o b u l g i n g by i t s e l f . 5.1.3 Bending/Unbending s t r a i n F i g s . 5 . 1 9 and 5 . 2 0 show t h e r e s u l t s of c a l c u l a t i o n s of t he b e n d i n g s t r a i n , E x , f o r Case 1. The r e m a i n i n g r e s u l t s a r e p r e s e n t e d i n Appendix V I . The major c h a r a c t e r i s t i c s of the r e s u l t s a r e as f o l l o w s . The bend i n g s t r a i n , e ^ , i s o b s e r v e d t o i n c r e a s e from one r o l l b e f o r e t h e b e n d i n g p o i n t t o R o l l No.1 beyond w h i c h i t r e a c h e s a s t e a d y - s t a t e l e v e l . The upper and lower s h e l l s deform about t h e i r own n e u t r a l axes t h e E CJ CNJ i-v. too.*; . -v. •".:• \ > 1 . 1 1 » u '.'•V s Fig. 5.8 Sulfur Print of a Longitudinal Section ; Rating of Internal Cracks - 0.2 46 E 46 F i g . 5.9 S u l f u r P r i n t of a L o n g i t u d i n a l S e c t i o n ; R a t i n g o f I n t e r n a l C r a c k s = 0 . 5 . Ln Fig. 5.10 Sulfur Print of a Longitudinal Section ; Rating of Internal Cracks =1.0. 46 ON F i g . 5.11 Predicted Bending and Bulging Strain, e x l n Case 1. (Upper Shell, V=1.6m/-min) 78 Roll 0 c |-0-5f CP c J ? CD c o c <u m No. at Straightening Zone 0 I 1 1 2 R = IO-5m 'R = 471 mm V = 1-6 m / m i n S = 8 3 mm T o =930°C Inner sur face — Outer surface -2-01 u C o s e I Fig. 5.12 Predicted Bending and Bulging Strain, e x i n Case 1. (Lower Shell, V=1.6m/min) 79 . C o s e 2 £ 1-01 c 'o I- CO c CD c o C P C c OJ CD 0-5 R = IO-5m l R= 471 mm V =1-2 m /min S =97mm T 0 = 900°C — Inner sur foce - - Outer s u r f a c e Roll No. at Straightening Zone Fig. 5.13 Predicted Bending and Bulging Strain, e i n Case 2. (Upper Shell, V=l.2m/min) 80 r Case 3 Fig. 5.14 Predicted Bending and Bulging Strain, ^ i n Case 3. (Upper Shell, V=l.Om/min) 81 Fig. 5.15 Predicted Bending and Bulging Strain, in Case 3. (Lower Shell, V=l.Om/min) F i g . 5.16 P r e d i c t e d D i s t o r t i o n Due t o B u l g i n g i n Case 1. (Lower S h e l l ) 00 XX-STRAIN 1 = - 0 . 4 2 5 7 E - 0 3 2 = - 0 . 2 5 5 6 E - 0 3 3 - -0.8544E-04 4 - 0.8471E-04 5 = 0 . 2 5 4 9 E - 0 3 6 - 0 . 4 2 5 0 E - 0 3 7 = 0 . 5 9 5 2 E - 0 3 8 = 0.7653E-03 9 = 0 .9355E-03 10 = 0 . 1 1 0 6 E - 0 2 Fig. 5 .17 Predicted XX-STRAIN Due to Bulging in Case 1. (Lower Shell) XY-STRAIN 1 = -0.4574E-03 2 = -0.3410E-03 3 = -0.2245E-03 4 = -0.1080E-03 5 = 0.8423E-05 6 = 0.1249E-03 7 = 0.2413E-03 8 = 0.3578E-03 9 - 0.A743E-03 10 = 0.5907E-03 Fi g . 5.18 Predicted XY-STRAIN Due to Bulging i n Case 1. (Lower Shell) 85 Table V Maximum bulging d e f l e c t i o n between No.1 and No.2 r o l l s , Bulging *B (mm) Bending and Bulging 6 T (mm) CASE 1. U 0.22 0.87 L 0.22 0.39 2. U 0 . 1 3 0.26 3. TJ 0.11 0.17 L 0.11 0.12 4. U 0.15 0.40 5. U 0 . 1 3 0.22 6 . U 0.08 0 . 1 6 7. U • 0.21 0.51 8. U 0.10 0.12 9. TJ 0.14 0.35 10. TJ 0.17 0.39 TJ : upper s h e l l , L : lower s h e l l 86 l o c a t i o n s of w h i c h a r e r o u g h l y s y m m e t r i c a l w i t h r e s p e c t t o t h e c e n t e r p l a n e of t h e s l a b t h i c k n e s s , 7 and 14 mm r e s p e c t i v e l y from t h e i n n e r s u r f a c e of each s h e l l as shown i n Figs.5. 1 9 and 5.20. T h e r e f o r e t h e b e n d i n g s t r a i n d i s t r i b u t i o n s of t h e upper and l o w e r s h e l l s a l s o a r e s y m m e t r i c a l w i t h r e s p e c t t o t h e c e n t e r p l a n e o f s l a b t h i c k n e s s . S m a l l peak s t r a i n s a r e o b s e r v e d between R o l l No.-1 and t h e t a n g e n t ( O ) r o l l a t t h e i n n e r s u r f a c e ; however t h i s peak can be a t t r i b u t e d t o the s i m p l i f y i n g a p p r o x i m a t i o n s of t h e upstream boundary c o n d i t i o n and the s i m u l a t i o n of a dynamic p r o c e s s by a t w o - s t a g e b e n d i n g model. To check t h e e f f e c t o f the b e n d i n g s i m u l a t i o n a t h r e e - s t a g e b e n d i n g model was r u n and the peak s t r a i n a t t h i s l o c a t i o n was found t o d e c r e a s e and t o move . f u r t h e r downstream. D e s p i t e t h e s e s i m p l i f i c a t i o n s , t h e s t r a i n and s t r e s s d i s t r i b u t i o n s at t h e b e n d i n g p o i n t and downstream of i t s h o u l d be r e a s o n a b l y a c c u r a t e owing t o t h e good - agreement t h a t was o b t a i n e d between the model p r e d i c t i o n s and t h e p l a n t .data on t h e o c c u r r e n c e of i n t e r n a l c r a c k s d e s c r i b e d e a r l i e r . F i g . 5 . 2 1 shows t h e p r e d i c t e d c u r v a t u r e , p , of t h e s t r a n d , where p was c a l c u l a t e d from t h e r e s u l t s of b e n d i n g s t r a i n as f o l l o w s ; i n which P = ( e 2 - z± ) /Ay e2 = s t r a i n a t o u t e r s u r f a c e z1 = s t r a i n a t i n n e r s u r f a c e ( 1 6 ) 87 Ay =distance between outer and inner surfaces As to the question of whether the strand is straightened along the r o l l p r o f i l e or not, the results show that bending occurs along the curvature determined by the r o l l p r o f i l e as shown in Fig.5.21. A similar finding has been reported by O n i s h i 3 1 for the one-dimensional dynamic analysis of bending of continuously cast slabs. Fig.5.22 shows the relationship between bending strain in the upper shell and r o l l pitch for different surface temperatures at the straightening point. Geometrical s t r a i n , which was calculated by assuming a neutral axis at the center plane of the slab thickness, also is shown on the same figure as a. broken line for comparison( y is the distance from the center plane). From the results, i t is evident that the bending strain i s independent of r o l l pitch and smaller than the above geometrical strain by about 0.3%. However, a small dependence can be seen on the surface temperature;the bending strain increases by 0.05% with a temperature increase of 90°C because the s t i f f n e s s of the strand is diminished so that the deformation i s localized around the bending point. Thus, the elongation due to bending is enhanced at the straightening zone at higher, temperatures. Figs'.5.23 and 5.24 show the relationship between bending strain and shell thickness for the 10.5m and 8.0m machine r a d i i . The bending strain distributions in the upper 88 a n d l o w e r s h e l l s a r e s e e n t o be s y m m e t r i c a l a n d t h e b e n d i n g s t r a i n a t t h e i n n e r s u r f a c e c h a n g e s l i n e a r l y w i t h an i n c r e a s e i n s h e l l t h i c k n e s s . A c c o r d i n g t o t h e s e r e s u l t s , t h e b e n d i n g s t r a i n a t t h e i n n e r s u r f a c e o f t h e u p p e r s h e l l c a n h a v e a n e g a t i v e v a l u e i f t h e s h e l l t h i c k n e s s i s l a r g e r t h a n 106 mm. F i g . 5 . 2 5 shows t h e i n f l u e n c e o f m a c h i n e r a d i u s ( c u r v a t u r e ) on b e n d i n g s t r a i n i n t h e u p p e r s h e l l . The b e n d i n g s t r a i n a t t h e o u t e r s u r f a c e i n c r e a s e s l i n e a r l y w i t h a c h a n g e o f c u r v a t u r e . H o w e v e r , t h e s t r a i n a t t h e i n n e r s u r f a c e i s n o t much i n f l u e n c e d by t h e c h a n g e , o f m a c h i n e r a d i u s . The r e a s o n i s t h a t t h e n e u t r a l a x i s o f b e n d i n g i s l o c a t e d v e r y c l o s e t o t h e i n n e r s u r f a c e a n d t h e r e f o r e t h e s t r a i n a t t h e i n n e r s u r f a c e i s a l m o s t i n d e p e n d e n t o f c u r v a t u r e . 5.2 C o r n e r s t r a i n a n d c r a c k f o r m a t i o n . A l t h o u g h t h i s a n a l y s i s h a s f o c u s e d on s t r e s s e s and s t r a i n s a t t h e l o n g i t u d i n a l m i d - p l a n e o f a s l a b i t i s p o s s i b l e t o e s t i m a t e s t r a i n a t t h e c o r n e r a s w e l l , b a s e d on i n s i g h t g a i n e d f r o m t h e t h r e e - d i m e n s i o n a l , e l a s t o - p l a s t i c c a l c u l a t i o n s . I f b u l g i n g i s n e g l e c t e d t h e s t r a i n d i s t r i b u t i o n i n t h e edge s h e l l i s a r e s u l t o f b e n d i n g w h i c h c a n be c a l c u l a t e d by c o n s i d e r i n g a n e u t r a l a x i s a t t h e c e n t e r o f t h e s l a b t h i c k n e s s . T a b l e V I p r e s e n t s t h e p r e d i c t e d b e n d i n g s t r a i n s a t t h e c o r n e r c l o s e t o t h e s o l i d i f i c a t i o n f r o n t a s w e l l a s b u l g i n g a n d b e n d i n g s t r a i n a t t h e i n n e r s u r f a c e o f t h e s h e l l i n t h e m i d - 89 1.0 x ty R = IO-5m l R= 47I mm V = l - 6 m / m i n S= 8 3 m m T0= 930°C C o s e i € 2 'o CO 0.5 c T J C CD 0 Inner s u r f a c e Ou le r s u r f a c e -1 Roll 0 1 No. at Straightening Zone ro €, =0-1 x i o _ 2 - ^ = 0 8 4 X 1 0 Fig. 5.19 Predicted Bending Strain, e x in Case 1. (Upper Shell, V=1.6m/min) 90 0| ty c 2 -Q5 to -2 V - 0 21X10 V - 0 - 8 2 X 1 0 I ^ neutrol axis No. at- Straightening Zone 0 1 Inner s u r f a c e — Outer su r face . CP c X J c o CD -1.0h R =10-5 m l R = 4 7 l m m V =1-6 m / m i n S = 8 3 m m T 0=-930°C v, — T~X.' ^ N / \ € 2 C o s e I L Fig. 5.20 Predicted Bending Strain, e x in Case 1. (Lower Shell, V=1.6m/min) 1 I ~1 7 Roll No. at Straightening Zone Fig. 5.21 Predicted Curvature of the Shell Due to Bending in Case 92 ~ U > r c o (/) cn c m outer (Y=1ia9) inner (Y=40.1) T(«c) • • 850 o • 900 A 990 cxxxTetricd strain 400 471 540 Roll Pitch (mm) Fig. 5.22 Relation between Bending Strain, e x and Roll Pitch Predicted by the Finite-Element Bending Analysis. _ J i i • 80 90 100 no Shell thickness (mm) Fig. 5.23 Relation between Bending Strain, and Shell Thickness Predicted by the Finite-Element Bending Analysis. (Machine Radius = 10-.5n>) 93 c CO cn c TJ C Q) CQ outer (Y=H8S) inner T Cc) o • 900 ^metrical strain 100 8 0 90 110 Shell thickness (mm) Fig. 5.24 Relation between Bending Strain, c x and Shell Thickness Predicted by the Finite-Element Bending Analysis. (Machine Radius = 8.0m) 13 - f— machine radius ( m ) 10.5 8 c a i_ TJ C at CO outer (Y=1183) inner (Y= 52.4) inner (Y=«X1) inner (Y=32.2) T Cc) o • • • 900 geometrical strain o 'A 'V 08 09 1.0 1.1 12 13 .-^ curvature p (Vmm) x10"4 Fig. 5.25 Relation between Bending Strain, e x and Machine Radius (Curvature) Predicted by the Finite-Element Bending Analysis, 94 p l a n e . F rom t h e r e s u l t s o f C a s e 2,4,5,6 a n d 7, l o w s u r f a c e t e m p e r a t u r e s a n d s m a l l r o l l p i t c h e s a r e p r e f e r a b l e t o p r e v e n t i n t e r n a l c r a c k s i n t h e m i d - p l a n e , s i n c e t h e s e c o n d i t i o n s s u p p r e s s t h e b u l g i n g s t r a i n . I n t h e c a s e o f an 8m m a c h i n e r a d i u s ( C a s e 8 , 9 ) , t h e t h r e s h o l d c a s t i n g s p e e d t o e n s u r e t h a t i n t e r n a l c r a c k s do n o t a p p e a r a t t h e c o r n e r i s 0 . 9 m / m i n ( b a s e d on t h e c r i t i c a l s t r a i n ) . F o r a 13m m a c h i n e r a d i u s ( C a s e 1 0 ) , t h e i n t e r n a l c r a c k s o c c u r i n t h e m i d - p l a n e due t o b u l g i n g , a n d t h e t h r e s h o l d c a s t i n g s p e e d i s n e a r l y 1.4m/min ( b a s e d on t h e c r i t i c a l s t r a i n ) . T h u s , i n one p o i n t b e n d i n g b o w - t y p e c a s t e r s , t h e s m a l l m a c h i n e r a d i u s o f 8.0m i s o b v i o u s l y u n f a v o r a b l e c o m p a r e d w i t h t h e v a l u e s o f 10.5m a n d 13m o w i n g t o t h e l o w c a s t i n g s p e e d a t w h i c h c r a c k s f o r m . 5.3 C r e e p e f f e c t s on t h e c r i t i c a l s t r a i n The c r i t i c a l s t r a i n f o r i n t e r n a l c r a c k s r e p o r t e d i n t h e l i t e r a t u r e e x h i b i t s some s c a t t e r ( 0 . 2 - 3 . 0 % a t a s t r a i n r a t e o f 1 x 1 0 " * s " 1 ) a s m e n t i o n e d p r e v i o u s l y i n S e c t i o n 2.1. I n t h e p r e s e n t s t u d y , t h e e s t i m a t e d v a l u e o f t h e c r i t i c a l s t r a i n i s 0 . 2 5 - 0 . 3 % a t a s t r a i n r a t e o f 3 x l 0 ~ * s " 1 b a s e d on t h e a p p e a r a n c e o f c r a c k s i n s l a b s . H o w e v e r , t h i s c r i t i c a l v a l u e may be an u n d e r e s t i m a t e s i n c e c r e e p h a s n o t f u l l y b e e n a c c o u n t e d f o r i n t h i s a n a l y s i s . C r e e p e f f e c t s h a v e been t a k e n i n t o a c c o u n t p a r t i a l l y b y c o n s i d e r i n g an a p p r o x i m a t e s t r a i n r a t e f o r t h e s t r e s s - s t r a i n c u r v e s . I f t h e m o d e l were a b l e t o c o n s i d e r c r e e p t h e b u l g i n g s t r a i n w o u l d be i n c r e a s e d a n d a s a r e s u l t t h e Table VI Bending and B u l g i n g s t r a i n at s o l i d i f i c a t i o n f r o n t . c e n t e r c o r n e r i n t e r n a l c r a c k s CASE 1. U 0.55/0.65 0.4 c e n t e r , c o r n e r L 0.1 -0.4 no c r a c k 2. U 0.25/0.3* 0.27 ( ^ c r i t i c a l s t r a i n ) 3. u 0.15/0.2 0.18 no c r a c k L 0.1/0.13 -0.18 no c r a c k 4. U 0.38/0.4 0.27 c e n t e r 5. U 0.15/0.23 0.27 no c r a c k 6. U 0.15/0.20 0.27 no c r a c k 7. U 0.3/0.4 0.27 c e n t e r 8. u 0.15/0.2 0.35 c o r n e r 9. u 0.35/0.4 0.53 c e n t e r , c o r n e r 10. u 0.3/0.33 0.21 c e n t e r U : upper s h e l l , L : lower s h e l l Ln 96 t o t a l s t r a i n of b e n d i n g and b u l g i n g s h o u l d be i n c r e a s e d . F i g . 5 . 2 6 shows the t o t a l s t r a i n , e x ,as a f u n c t i o n of b u l g i n g s t r a i n , E ,and bend i n g s t r a i n , E . From the r e s u l t s , the B u c o r r e l a t i o n among t h e s e v a r i a b l e s i s as f o l l o w s ; e T = ( 2 - 5 ) e B + ^ ( 1 7 ) As i s a p p a r e n t from E q . ( l 7 ) , b u l g i n g s t r a i n a f f e c t s the t o t a l s t r a i n s i g n i f i c a n t l y and hence c r e e p e f f e c t s on b u l g i n g s h o u l d be c o n s i d e r e d t o d e t e r m i n e t h e c r i t i c a l s t r a i n more p r e c i s e l y . I f , f o r i n s t a n c e , b u l g i n g s t r a i n i s i n c r e a s e d from 0.06% t o 0.12% a t the bend i n g s t r a i n of 0 . 1 % which i s a c o n d i t i o n of Case 2, the c r i t i c a l s t r a i n w i l l be i n c r e a s e d e a s i l y from 0.25% t o 6 . 6 % , see F i g . 5 . 2 6 . Bend i ng St ra i n (%) F i g . 5.26 P r e d i c t e d T o t a l B e n d i n g and B u l g i n g S t r a i n , , a t an I n n e r S u r f a c e as a F u n c t i o n of B u l g i n g S t r a i n , eu, and B e n d i n g S t r a i n , E U . 98 Chapter 6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 6. 1 Conclusions A two-dimensional e l a s t o - p l a s t i c model has been developed, based on the plane-stress finite-element method to cal c u l a t e the bending and bulging deformation of p a r t i a l l y s o l i d i f i e d continuously cast s t e e l slabs during straightening. The findings of the study are as follows; (1) The model predictions of int e r n a l cracks showed good agreement with the plant data of Oita works, NSC. The int e r n a l cracks are predicted to occur mainly in the upper s h e l l beneath the r o l l support points in the straightening zone based on a c r i t i c a l cracking s t r a i n of 0.25 to 0.3% at a s t r a i n rate of 1x10-" s- 1 . (2) The upper and lower s h e l l s deform independently about t h e i r respective neutral axes. 99 ( 3 ) The r o l l f r i c t i o n f o r c e p l a y s t h e most i m p o r t a n t r o l e i n t h e b e n d i n g a n a l y s i s a n d a v a l u e o f 0.45 was a d o p t e d f o r t h e c o e f f i c i e n t o f r o l l f r i c t i o n . O w i n g t o t h e r e s t r a i n t e x e r t e d on t h e s t r a n d by t h e r o l l f r i c t i o n f o r c e t h e n e u t r a l a x e s o f t h e two s h e l l s s h i f t i n w a r d , p a s t t h e s o l i d i f i c a t i o n f r o n t i n t o t h e m o l t e n s t e e l , a n d a s a r e s u l t t h e u p p e r and l o w e r s h e l l s d e f o r m l i k e a s i n g l e beam. The n e u t r a l a x e s o f t h e u p p e r and l o w e r s h e l l s a r e l o c a t e d v e r y c l o s e . t o t h e i r r e s p e c t i v e s o l i d i f i c a t i o n f r o n t s . ( 4 ) The s t r a i n d i s t r i b u t i o n o f e x i s l i n e a r t h r o u g h t h e s h e l l t h i c k n e s s a n d h e n c e b e n d i n g s t r a i n , , f o l l o w s t h e o r d i n a r y b e n d i n g beam t h e o r y . w h e r e y i s t h e d i s t a n c e f r o m t h e n e u t r a l a x i s a n d R i s t h e b e n d i n g r a d i u s . ( 5 ) B e n d i n g o c c u r s a l o n g t h e c u r v a t u r e d e t e r m i n e d by t h e r o l l p r o f i l e . ( 6 ) The b e n d i n g s t r a i n d e p e n d s s l i g h t l y on t h e s u r f a c e t e m p e r a t u r e o f t h e s t r a n d i n c r e a s i n g by 0.05% w i t h a t e m p e r a t u r e i n c r e a s e o f 90°C. 100 (7) The bulging' d e f l e c t i o n i s enhanced s i g n i f i c a n t l y as a c resu l t of i n t e r a c t i o n with bending. The resu l tant bulging de f l ec t ions are greater in the upper than in the lower s h e l l . (8) The shear s t r a i n , e x y , i s comparable to the e x component in the case of the bulging a n a l y s i s , whereas e i s f a i r l y smal l , c lose to one fourth of e j X component, in the case of the combined bending and bulging a n a l y s i s . (9) The t o t a l s t r a i n , e T , can be expressed in terms of each component of bulging s t r a i n , E b , and bending s t r a i n , e u , as fo l lows . E t = (2 - 5) e B +. c u The bulging s t r a i n a f fec t s the t o t a l s t r a i n s i g n i f i c a n t l y and hence to prevent i n t e r n a l cracks i t i s important to suppress the bulging by ensuring low surface temperatures and have a small r o l l p i t c h in the s t ra ighten ing zone. (10) The predic ted c r i t i c a l s t r a i n for i n t e r n a l cracks i s 0.25 - 0.30% at 1x10-" s"1 for low-carbon s t e e l s . However, i t w i l l be necessary to take into account 101 c r e e p e f f e c t s to o b t a i n a more p r e c i s e v a l u e of the c r i t i c a l s t r a i n . (11) In o n e - p o i n t b e n d i n g , b o w - t y p e c a s t e r s , a s m a l l machine r a d i u s of 8m i s o b v i o u s l y u n f a v o r a b l e compared w i t h the v a l u e s of 10.5m and 13m because a t normal c a s t i n g speeds the t e n s i l e s t r a i n : a t the s o l i d i f i c a t i o n f r o n t exceeds the c r i t i c a l v a l u e for c r a c k f o r m a t i o n . 102 6.2 S u g g e s t i o n s f o r f u t u r e work An i m p o r t a n t d i r e c t i o n f o r f u r t h e r r e s e a r c h i s t h e e x p e r i m e n t a l measurement of s e v e r a l p a r a m e t e r s a d o p t e d i n t h e p r e s e n t model such as the c o e f f i c i e n t of r o l l f r i c t i o n and t h e c r i t i c a l s t r a i n f o r i n t e r n a l c r a c k s . The r e s u l t s of s u c h an i n v e s t i g a t i o n would h e l p c o n c l u s i v e l y e s t a b l i s h the v a l i d i t y of the p r o p o s e d model. 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H i r o m o t o : " A n a l y s i s o f B u l g i n g i n C o n t i n u o u s l y C a s t S l a b s b y t h e C r e e p M o d e l " , T e t s u - t o - H a g a n e , V o l . 6 7 , 1 9 8 1 , p p 1 1 7 2 - 1 1 7 9 25-. T . M a t s u m i y a a n d Y .Nakamura: T e t s u - t o - H a g a n e , V o l . 6 8 , 1 982,A145 ' 26. M o r i t a : K o b e - s e i k o - g i h o , 29-3 ,1979, p p 5 5 - 5 9 27. K . W u n n e n b e r g : S t a h l u. E i s e n , 98 , 1978, p p 2 5 4 - 2 5 9 28. 0 . C . Z i e n k i e w i c z : " The F i n i t e E l e m e n t M e t h o d i n E n g i n e e r i n g S c i e n c e " , New Y o r k , M c G r a w - H i l l , 1974 29. Y.Yamada : " R e c e n t a d v a n c e s i n m a t r i x m e t h o d s o f s t r u c t u r a l a n a l y s i s a n d d e s i g n " , 1969 30. R . H i l l : " The t h e o r y o f c o m b i n e d p l a s t i c and e l a s t i c d e f o r m a t i o n , w i t h p a r t i c u l a r r e f e r e n c e t o a t h i c k t u b e u n d e r i n t e r n a l p r e s s u r e " , P r o c e e d i n g s o f t h e R o y a l S o c i e t y , A . V o l . 1 9 1 , 1947, p p 2 7 8 - 3 0 3 106 3 1 . K . O n i s h i , K . N a g a i a n d M . W a k a b a y a s h i : " A n u m e r i c a l a n a l y s i s o f s t r a i n s i n s l a b s a n d f o r c e s on r o l l e r s i n t h e s t r a i g h t e n i n g z o n e o f c o n t i n u o u s c a s t i n g m a c h i n e " , T e t s u - t o - H a g a n e , V o l . 6 7 , 1 9 8 1 , p p l 1 6 2 - 1 1 7 1 32. J . A . S c h e y : " M e t a l d e f o r m a t i o n p r o c e s s e s ( F r i c t i o n and L u b r i c a t i o n ) " , MARCEL DEKKER, New Y o r k , 1970 33 . S u z u k i : T r a n s . J a p a n I n s t i t u t e o f M e t a l s , 32, 1968, p p 1 3 0 l 34. T . I n o u e a n d H . T a n a k a : " P r o g r e s s i n l a r g e - s e c t i o n s l a b c o n t i n u o u s c a s t i n g t e c h n i q u e s a t N i p p o n S t e e l C o r p o r a t i o n " , N i p p o n S t e e l T e c h n i c a l R e p o r t , No.13, J u n e 1979 ,pp1-23 35 . N . Y a m a u c h i , H . M i s u m i , Y . U c h i d a a n d T.Yamamoto: " I n t e r n a l C r a c k s i n C o n t i n u o u s l y C a s t S l a b s " , N i p p o n S t e e l T e c h n i c a l R e p o r t , N o.13, J u n e 1979, p p 6 2 - 7 2 36. S . S . D a n i e l :" R o l l C o n t a i n m e n t M o d e l f o r S t r a n d - C a s t S l a b - a n d B l o o m s " , 2nd P r o c e s s T e c h n o l o g y C o n f . , C h i c a g o , 1981, P P 1 0 2 - 1 1 3 37. K . N a r i t a , T . M o r i a n d J . M i y a z a k i : " E f f e c t o f D e f o r m a t i o n on t h e F o r m a t i o n o f I n t e r n a l C r a c k s i n C o n t i n u o u s l y C a s t B l o o m s " , T e t s u - t o - H a g a n e , V o l . 6 7 , 1981, p p l 3 0 7 - 1 3 l 6 3 8 . H . S u z u k i : " C h a r a c t e r i s t i c s o f E m b r i t t l e m e n t i n S t e e l s a b o v e 600°C", T e t s u - t o - H a g a n e , V o l . 6 5 , 1979,S2038 3 9 . T . M a t s u m i y a : T e t s u - t o - H a g a n e , V o l . 6 9 , 1983, S169 40 . T . O b i n a t a : " G e n e r a l V i e w o f t h e C o n t i n u o u s C a s t i n g E q u i p m e n t " , T e t s u - t o - H a g a n e , V o l . 6 0 , 1974, p p 7 4 1 - 7 5 4 4 1 . F . W e i n b e r g : " The D u c t i l i t y o f C o n t i n u o u s l y C a s t S t e e l N e a r t h e M e l t i n g P o i n t - H o t T e a r i n g " , M e t a l . T r a n s . B , V o l . l O B ,June 1979, p p 2 l 9 - 2 2 7 107 4 2 . H . S u z u k i , S . N i s h i m u r a a n d S . Y a m a g u c h i : " C h a r a c t e r i s t i c s o f h o t d u c t i l i t y i n s t e e l s s u b j e c t e d t o t h e m e l t i n g a n d s o l i d i f i c a t i o n " , T r a n s . I S I J , V o l . 2 2 , 1982, p p 4 8 - 5 6 43 . G.Komma : " D e s i g n a n d O p e r a t i o n a l A s p e c t s i n C o n t i n u o u s C a s t i n g o f W i de S l a b s " , I r o n a n d S t e e l E n g . , J u n e 1 9 7 3 , p p 6 8 - 7 3 44. A . V a t e r l a u s :" F i n i t e e l e m e n t a n a l y s i s f o r s l a b s t r a i g h t e n i n g w i t h l i q u i d c o r e " , T e t s u - t o - H a g a n e , V o l . 6 9 , 1982,S170 45 . S . N a g a t a a n d K . Y a s u d a : T e t s u - t o - H a g a n e , V o l . 6 8 , 1982, S991 46. H . M i s u m i : P r i v a t e C o m m u n i c a t i o n 1 0 8 APPENDIX I MECHANICAL PROPERTIES ADOPTED IN THE BULGING CALCULATION FOR THE COMPARISON WITH THE EXPERIMENTAL RESULTS OF Morita Material 1 Ma t e r i a1 2 Mater i a l 3 Ma t e r i a 1 4 T C O 1101 1 205 1 309 1413 E MPa 17738 1 5827 13916 1 01 23 ay MPa 14.8 10.1 6.3 2.4 0.36 0.37 0. 38 0.39 a= 7 5 . 4 e ° ' 2 3 o= 4 7 . O e 0 ' 2 1 <r- 2 1 . 5 e 0 ' 1 6 r n „ 0 .11 a = 6.0 e 109 APPENDIX I I DERIVATION OF THE FINITE-ELEMENT EQUATIONS FOR THE ELASTO-PLASTIC PROBLEMS D i s p l a c e m e n t s { 5 } w i t h i n each element a re g i v e n by {6} = [N] {6 } e (A.V) where [ N ] i s a m a t r i x of shape f u n c t i o n s and {6 f i s a v e c t o r of n o d a l d i s p l a c e m e n t s . The s t r a i n { e } and s t r e s s { a } i n t h e element are g i v e n by {e} = [B] {6 } e io] = [D] ie] (A.2) By a p p l y i n g the p r i n c i p l e of v i r t u a l work t o t h e s e e l e m e n t s and summing the i n d i v i d u a l e q u i l i b r i u m e q u a t i o n s f o r a l l e l e m e n t s , Eq.(A.3) i s o b t a i n e d . The n o d a l f o r c e s , d i s p l a c e m e n t s and t h e d i s t r i b u t e d l o a d s { p } can nov? be r e l a t e d t h r o u g h 110 i F } = [ K ] { } + { Fp' } ( A . 3 ) w h e re and [ K ) = Z / [ B ] T l D ] l B ] d (vol) {• Fp }= - I f I "N J T { P } d (vol) E - summation over a l l elements. E q u a t i o n (A.3) c a n be s o l v e d f o r t h e d i s p l a c e m e n t s . { 6 } = [ K J - 1 C { F } - { Fp } ) ( A . 4 ) F i n a l l y , u s i n g E q . ( A . 2 ) , one c a n c a l c u l a t e t h e s t r a i n a n d s t r e s s d i s t r i b u t i o n o v e r t h e e n t i r e r e g i o n . a n a l y s i s i s more c o m p l i c a t e d s i n c e t h e [ D ] a n d [ K ] m a t r i c e s become s t r a i n ( o r s t r e s s ) d e p e n d e n t . T h e r e f o r e E q . ( A . 3 ) i s s o l v e d by an i n c r e m e n t a l m e t h o d . 2 9 The l o a d i s a p p l i e d i n c r e m e n t a l l y and t h e [ K ] m a t r i x i s a d j u s t e d a f t e r e v e r y i n c r e m e n t . The l o a d i n c r e m e n t s a r e a d j u s t e d , s u c h t h a t w i t h e a c h i n c r e m e n t 30 e l e m e n t s y i e l d . A f t e r a l l e l e m e n t s y i e l d , t h e r e m a i n i n g l o a d i s d i v i d e d i n t o s e v e r a l e q u a l i n c r e m e n t s . N o m e n c l a t u r e ; [ ] m a t r i c e s { } v e c t o r s U n d e r c o n d i t i o n s o f p l a s t i c d e f o r m a t i o n t h e s t r e s s ** d e t a i l s o f t h e d e r i v a t i o n s a r e d i s c u s s e d b y Z i e n k i e w i c z 2 8 a n d Y a m a d a 2 9 . I l l APPENDIX I I I MATERIAL MATRIX[D](plane s t r e s s ) USED IN THE FINITE ELEMENT Below the y i e l d p o i n t , the e l a s t i c m a t r i x [ D e ] f o r th e p l a n e s t r e s s c o n d i t i o n i s g i v e n by 1 - v 2 I . v 0 1 0 sym 1-y (B. 1 ) where E i s the Young's Modulus and v i s P o i s s o n ' s r a t i o . Under c o n d i t i o n s of p l a s t i c d e f o r m a t i o n , t h e p l a s t i c m a t r i x [ D P ] w h i c h was d e v e l o p e d by Yamada 1 9 f o r a M i s e s m a t e r i a l i s as f o l l o w s . I DP ] 1 - 1 v 1 sym 0 0 1-v S1 S2 S s2 S1 S6 S2 S6 sym (B.2) 112 W h e r e S = j a % . + S.a; + s / y + 2S6x'xy a n d  S l = 1~2 ( <r'x + vo y ) . S 2- ^ va'x + a ; ) . S,- "a and 7 P a r e the e f f e c t i v e s t r e s s and s t r a i n and H* i s t h e s l o p e of t h e o, eP c u r v e . ox'., o y t a n d x x y a r e d e v i a t o r i c s t r e s s e s . U n l o a d i n g check has been p e r f o r m e d by c a l c u l a t i n g de p as f o l l o w s . S,d e + S.de + S,dy ,_ , , d £ P 1 x 2 y 6 lxy (B.3) 3 J_ 2 o" i f de p <0 u n l o a d i n g Once u n l o a d i n g o c c u r s , t h e m a t e r i a l m a t r i x of the ele m e n t i s changed from [ D p ] t o [ D e ] . 113 APPENDIX IV THICK-WALLED CYLINDER UNDER INTERNAL PRESSURE(plane s t r a i n ) p e r f o r m e d f o r the case of a t h i c k - w a l l e d c y l i n d e r under i n t e r n a l p r e s s u r e f o r w h i c h an a n a l y t i c a l s o l u t i o n of the s t r e s s f i e l d i s a v a i l a b l e . The geometry of the c y l i n d e r a<r<b i s shown i n F i g . I V . 1 . The e l a s t o - p l a s t i c boundary i s l o c a t e d a t r=c. The d i s p l a c e m e n t s i n the 6 d i r e c t i o n on the r a d i a l b o u n d a r i e s were c o n s t r a i n e d due t o the a x i a l symmetry. The d i m e n s i o n s of the c y l i n d e r were s e l e c t e d as b=2a, i n o r d e r t o compare the n u m e r i c a l r e s u l t s w i t h the a n a l y t i c a l s o l u t i o n of H i l l 3 0 . M e c h a n i c a l p r o p e r t i e s of t h e m a t e r i a l were as f o l l o w s : P o i s s o n ' s r a t i o v-0.3 Shear modulus G = 4 x l 0 6 p s i Young's modulus E=10.4X10 6 p s i To check the a c c u r a c y of "EPIC-IV ?! a c a l c u l a t i o n was Y i e l d s t r e s s 114 The s t r e s s ^ z d e p e n d s on the s t r a i n h i s t o r y and must be o b t a i n e d from t h e P r a n d t l - R e u s s e q u a t i o n s . F i g . I V . 2 shows the c o m p a r i s o n of t h e c a l c u l a t e d c r z i n d i m e n s i o n l e s s form w i t h t h o s e of H i l l . E x c e l l e n t agreement i s o b s e r v e d . 115 Fig. IV.1 Geometry of a Thick Walled Cylinder, 0.4 Fig. IV.2 Comparison of the Calculated Stresses °z: Csolid points and lines) with Those Obtained by H i l l 3 0 " , 1 17 APPENDIX V ESTIMATION OF ROLL FRICTION FORCE IN CASE 1(UPPER SHELL) R o l l No. R o l l P i t c h mm R o l l F r i c t i o n Force K F i Cumulat ive R o l l F r i c t i o n Force H i F i S t ress MPa o i Average S t r a i n % e i E l o n g a t i o n min .At. i 11 435 133.2 0 0 0 0 10 435 136.2 133.2 6.4 0.008 0.03 9 435 137.2 269.4 12.9 0.02 0 .08 8 435 140.1 406.6 19.6 0.03 0.13 7 435 141.1 546.7 26.3 0.035 0.15 6 435 143.1 689.8 33.2 0.06 0.26 5 435 143.1 832.9 40.1 0.095 0.41 • 4 435 148.9 981.8* 47.3 0.15 0.65 3 471 157.7 1139.5 54.9 0.32 1.51 2 471 157.7 1297.2 62.5 0 .48 2.28 (B.P.) 1 471 157.7 1454.9 70.1 0.71 3.36 T o t a l 8.86 * Fu » 981.8 N is adopted as the force boundary condition on upstream edge. CFig.3. 13) APPENDIX VI RESULTS OF CALCULATION OF BENDING AND BULGING (CASE 2 TO CASE 10) 119 1.0 c c '•XD c CD 0 Cose 2 R = IO-5m ! R = 4 7 l m m V =l-2 m /min S =97 mm T =900°C Inner su r face cn Outer s u r f a c e a. 0.5 _1 0 1 Roll No. at- Straightening Zone V \ €2 Fig. VI.1 Predicted Bending Strain, e x in Case 2 . (Upper Shell) 120 r Cose 3 1.0 ty c "o CO c C CD 0.5 Oi R = IO-5m ' R = 4 7 I mm V = IO m / m i n s = I 0 6 mm V 9 0 0 ° C 1 » J . ^7 Inner sur foce Outer su r foce -1 0 ^1 Roll No. at- Straightening Zone €2 Fig. VI.2 Predicted Sending Strain, i n Case.3. (Upper Shell) 121 €,=-0-02 X I O " 2 € = ̂ 0 - 8 5 X 1 0 " cn Roll No. at- Straightening Zone A 0 1 2 x .E -0.5| O CO I n n e r s u r f o c e O u l e r s u r f o c e cr> c c OJ ca -1.0 R=IO-5m I =471 mm R V =I0 m / m i n S =106 mm T = 900 °C €2 Cose 3 Fig, VI.3 Predicted Bending Strain, c x in Case 3. (Lower Shell) Roll No. at Straightening Zone Fig. VI.4 Predicted Curvature of the Shell Due to Bending in Case 3. 1.0 1 1 1 1 R= 10-5 m 1 = 471 mm R V=l-2m/min * r "* S = 97 mm / / T0= 900°C / 1 1 r Cose 4 ty c o CO 0.5 c c CD o 1 Inner surfoce Outer surfoce -1 0 1 Roll No. at- Straightening Zone L O neutral axis €, =oi x i o _ 2 0-91 XI0 Fig. VI.5 Predicted Bending Strain, e x.in Case 4, (Upper Shell) 124 10 c o CO CP c '^0 5 CD TJ c D cn c T J C O) CD R = IO-5m ?= 471 mm V = 1-2 m /m in S = 97 mm T0 = 990°C r Cose 4 Inner surface Outer sur foce Roll No. at Straightening Zone Fig. VI.6 Predicted Bending and Bulging Strain, e x in Case 4. (Upper Shell) 125 T Cose 5 1.0 c 'o 00 CV cz o CD OS o l R= tO 5 m I = 471 mm V = l - 2 m / m i n S= 97 m m T̂ = 900°C -^-- ' -v €2 Inner su r foce Ou le r s u r f o c e -1 0 1 Roll No. at- Straightening Zone CO CNJ € =003XIC[J 0:79X10 Fig. VI.7 Predicted Bending Strain, ^ in Case 5, (Upper Shell) 126 Cose 5 i o h c "o CO C P c 3 CO •fO-5 -o c o c c cu CD R =10 5 m l R = 4 7 l m m V = 1-2 m /min S = 9 7 mm T 0 =850°C Inner sur face Outer sur face - 1 . 0 I Roll No. at Straightening Zone F i g . V I . 8 Predicted Bending and Bulging Strain, in Case 5 . (Upper Shell) 127 1.0 c c c CD GQ 0 Cose 6 R= IO-5m I =400mm R V=l -2m/ rn in S= 9 7 m m T0= 900°C - _ I— _ — _ ' k- I. — Inner s u r f o c e CO — Outer s u r f o c e -1 0 1 Rol l No. at Straightening Zone OD rsi € =0-09X10 2 €2= 0 8 4 X 1 0 Fig. VI.9 Predicted Bending Strain, £ x in Case 6. (Upper Shell) 128 c o CO C P c C P 3 0-5 CO T J c D CP C T J c OJ CD R = tO-5 m l R= 4 0 0 m m V=l 2 m/min S = 9 7 m m T 0 = 900°C r C o s e 6 / \ Inner su r face Outer s u r f a c e / -I 0 I Roll No. ot Straightening Zone V €2 Fig. VI.10 Predicted Bending and Bulging Strain, in Case 6. (Upper Shell) 129 1.0 c 'o CO |» 0.5 T J C 0) CO. ••"' 1 r - 1 1 Cose 7 R = 10-5 m - l R = 5 4 0 mm V = 1-2 m/min S = 9 7 m m T 0 =900 °C ** \ ' 1 - / — — f—«T ~~> ~ / \ ' \ € 2 . / / - s — Inner s u r f o c e t / — Outer s u r f o c e t - / 1 1 t t 1 1 1 / / ^ f e i - ' 1 0 1 No. at- Straightening Zone Fig. VI.11 Predicted Bending Strain, t in Case 7. (Upper Shell) 130 0 c 'o CO m 0-5 •o c o cn c •o c a> CD 0 cn Inner surfoce c Outer sur foce I Cose 7 -\ R = 10-5 m 'R = 5 4 0 mm V = 1-2 m/min S = 9 7 m m V 900°C A / \ / \ / \ Roll No. at Straightening Zone e 2 Fig. VI.12 Predicted Bending and Bulging Strain, i n Case 7, (Upper Shell) 131 1.0 c 'o cn c T> c OJ m R = 8 0 m l R = 47I mm V =0-9m/min S =97 mm TQ = 900°C Inner surfoce Outer surface 0.5h 0' Cose 8 -1 0 ^ 1 Roil No. at- Straightening Zone el Fig. VI.13 Predicted Bending Strain, e x in Case 8. (Upper Shell) 10 c o CO C P c f 0-51 X ) • c o CP c T J c cu CQ 0 R = 80m I =471 mm n V = 0-9 m/min S = 97 mm TQ = 900°C •Inner surfoce Outer surface Roll No. at Straightening Zone Fig. VI.14 Predicted Bending and Bulging Strain, i n Case 8. (Upper Shell) 133 Cose 9 10 R = 8 0 m ' R = 471 mm V = 1-2 m/min S = 8 3 m m V 900°C X Vl/ o CO 0-5 CP C T J C CU CO 0' Inner sur foce Outer sur foce Roll No. at- Straightening Zone €, =0-l9Xl6~f2 £,= 1 0 5 X 1 0 Fig, VI.15 Predicted Bending Strain, e x in Case 9. (Upper Shell) 134 10 co CP c CD •o cr o CP c c cu CD 0 5 0 c Inner sur foce o Outer s u r f o c e — r Cose 9 R =80m ^ =471 mm V = l-.2m/min S =83mm T0 = 900°C l €2 Roll No. ot Straightening Zone Fig. VI.16 Predicted Bending and Bulging Strain, in Case 9. (Upper Shell) 135 ty c o CO "D C OJ m r~ C o s e 10 R = l 3 0 m L ^ A T I m m V = l - 4 7 m / m i n S = 9 7 m m T 0=900°C X \ €2 — Inner surfoce • - Outer surfoce Roll No. at Straightening Zone Fig. VI.17 Predicted Bending Strain, e x in Case 10. (Upper Shell) 136 10 ' o TJ c o c n c T J c cu CQ , — Cose 10 J R = l 3 0 m l R = 4 7 l m m V = l - 4 7 m / m i n S = 9 7 m m T 0 =900°C / 00 Inner sur face / 1 1 ' cn - Outer surfoce Bu lg in  O  1 1 1 1 1 -I 0 I Roll No. at Straightening Zone \ \ €2 Fig. VI.18 Predicted Bending and Bulging Strain, e x in Case 10. (Upper Shell)

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